UK prime minister Theresa May said the UK will commit to maintaining the common travel area with Ireland.
Outlining her Brexit plan, Mrs May said the UK government will "make it a priority to deliver a practical solution” to the question of the shared land with the Irish State.
“Nobody wants to return to the borders of the past,” she said.
“The family ties and bonds of affection that unite our two countries mean that there will always be a special relationship between us."
She added that maintaining the common travel area with Ireland would be “an important part of the talks”.
The Irish Government welcomed this saying that for Ireland, "the priorities for the negotiation process that lies ahead are unchanged: our economic and trading arrangements, the Northern Ireland Peace Process including border issues, the common travel area, and the future of the European Union".
The Government said it "is under no illusion about the nature and scale of the Brexit challenge... But it is ready".
In her speech, Mrs May said that Britain will leave the European single market when it quits the European Union.
Mrs May said that her plans for Brexit cannot allow continued membership of the single market, which would require free movement of people and accepting the jurisdiction of the European Court of Justice.
Instead, she said that she will seek "the greatest possible access to the single market on a reciprocal basis, through a comprehensive trade agreement".
Mrs May said that she wanted to remain part of a customs agreement with the remaining 27 EU states, but said she had an "open mind" over whether this would be through associate membership of the Customs Union or through some other arrangement.
Her announcement came in a high-profile speech in London setting out her objectives for post-Brexit Britain.
She also revealed that the final Brexit deal reached between the UK and European Union will be put to a vote of both Houses of Parliament.
Mrs May did not make clear whether a vote against the agreement would result in the UK remaining in the EU or in Britain crashing out of the 28-nation bloc without a deal.
Speaking at Lancaster House, Mrs May said: "When it comes to Parliament, there is one ... way in which I would like to provide certainty. I can confirm today that the Government will put the final deal that is agreed between the UK and the EU to a vote in both Houses of Parliament before it comes into force."
Under Article 50 of the EU treaties, Britain will have two years to negotiate a deal after it informs the European Council of its intention to quit - something which Mrs May has said she will do by the end of March.
Chief EU negotiator Michel Barnier has suggested that an agreement must be concluded by October 2018 to allow time for ratification before Britain leaves in March 2019, meaning that the Commons and Lords votes are likely to come during that six-month period.
Mrs May said that Britain will maintain "practical arrangements on law enforcement and the sharing of intelligence material" with its former EU partners, as well as continuing to work as closely with European allies on foreign and defence policy as it now does with the EU.
Mrs May said the UK would regain control of its borders.
"We will get control of the number of people coming to Britain from the EU. Because, while controlled immigration can bring great benefits, filling skill shortages, delivering public services, making British businesses the world beaters they often are, when the numbers get too high, public support in the system falters."
The PM said Britain would no longer be under the jurisdiction of the European Court of Justice after Brexit.
Mrs May said remaining in the single market would mean "to all intents and purposes" not leaving the EU.
"As a priority we will pursue a bold and ambitious free trade agreement with the EU.
"This agreement should allow for the freest possible trade in goods and services between Britain and the EU's member states.
"It should give British companies the maximum possible freedom to trade with and operate within European markets and let European businesses do the same in Britain.
"But I want to be clear: what I am proposing cannot mean membership of the single market."
European leaders had stressed that single market membership meant accepting free movement of goods, services and people.
Being out of the EU but remaining in the single market would also mean being bound by the rules and regulations without having a say in how they are drawn up.
"It would, to all intents and purposes mean not leaving the EU at all," she said.
"That is why both sides in the referendum campaign made it clear that a vote to leave the EU would be a vote to leave the single market."
Mrs May said the UK would seek the "greatest possible access" to the single market through a "new, comprehensive, bold and ambitious free trade agreement".
The deal could take in elements of single market arrangements in areas such as the automotive industry or financial services, she said.
"An important part of the new strategic partnership we seek with the EU will be the pursuit of the greatest possible access to the single market on a fully reciprocal basis through a comprehensive free trade agreement," she said.
The Prime Minister added that leaving the single market would mean no longer having to pay "huge sums" to the EU budget - but the UK could continue to pay to participate in some programmes.
Mrs May repeated her desire to reach an early deal on the status of British citizens in the EU and those from the continent living in Britain.
She also vowed to ensure workers' rights are "fully protected and maintained".
Mrs May warned the EU against trying to "punish" the UK as a warning to others who might seek to leave the bloc.
She said seeking a punitive Brexit deal with the UK would be "an act of calamitous self-harm", adding that "no deal for Britain is better than a bad deal for Britain".
Mrs May said she did not want a long transitional period to move towards Britain's new relationship with the EU.
But she said that once a deal is agreed, it will be implemented with a "phased approach, delivering a smooth and orderly Brexit".
She said: "We will seek to avoid a disruptive cliff edge and we will do everything we can to phase in the new arrangements we require as Britain and the EU move towards our new partnership."
Listing her key objectives for the Brexit negotiations, Mrs May said: "These are the objectives we have set:
"Certainty wherever possible. Control of our own laws. Strengthening the United Kingdom.
"Maintaining the common travel area with Ireland. Control of immigration. Rights for EU nationals in Britain and British nationals in the EU. Enhancing rights for workers.
"Free trade with European markets. New trade agreements with other countries. A leading role in science and innovation. Co-operation on crime, terrorism and foreign affairs. And a phased approach, delivering a smooth and orderly Brexit."
And she added: "This is the framework of a deal which will herald a new partnership between the UK and the EU. It is a comprehensive and carefully considered plan that focuses on the ends, not just the means, with its eyes fixed firmly on the future and on the kind of country we will be once we leave."
On the customs union, Mrs May said she wanted a new relationship in order to allow the UK to strike trade deals around the world - and highlighted Donald Trump's comments on a possible agreement with the UK.
"Countries including China, Brazil and the Gulf states have already expressed interest in striking trade deals with us," she said.
"We have started discussions on future trade ties with countries like Australia, New Zealand and India. And President-elect Trump has said that Britain is not at the back of the queue for a trade deal with the United States, the world's biggest economy, but front of the line."
Mrs May said full customs union membership would prevent the UK from striking its own comprehensive trade deals.
"I want Britain to be able to negotiate its own trade agreements but I also want tariff-free trade with Europe, and cross-border trade there to be as frictionless as possible.
"That means I do not want Britain to be part of the common commercial policy and I do not want us to be bound by the common external tariff.
"These are the elements of the customs union that prevent us from striking our own comprehensive trade agreements with other countries. But I do want us to have a customs agreement with the EU."
She said she had an open mind whether that meant a "completely new" agreement, becoming "an associate member of the customs union in some way" or "remaining a signatory to some elements of it".
But she said: "I want to remove as many barriers to trade as possible and I want Britain to be free to establish our own tariff schedules at the World Trade Organisation, meaning we can reach new trade agreements not just with the EU but with old friends and new allies from outside Europe too."
Mrs May insisted that she was "anticipating success" in the Brexit talks, as she believed that the deep connections between Britain and its European neighbours meant that it would be in the EU's interests to strike a favourable deal.
"I do not believe that the EU's leaders will seriously tell German exporters, French farmers, Spanish fishermen, the young unemployed of the eurozone and millions of others that they want to make them poorer just to punish Britain and make a political point," she said.
"For all these reasons, and because of our shared values and the spirit of goodwill that exists on both sides, I am confident that we will follow a better path. I am confident that a positive agreement can be reached.
"It's right that the Government should prepare for every eventuality, but to do so in the knowledge that a constructive and optimistic approach to the negotiations to come is in the best interests of Europe and the best interests of Britain."
Mrs May said she believed that Britain's prospects in the Brexit negotiations were bolstered by "the strength and support of 65 million people willing us to make it happen"
She acknowledged: "After the division and discord, the country is coming together. The referendum was divisive at times and those divisions have taken time to heal."
But she insisted that the strength of Britain's democracy ensured that "when a vote has been held, we all respect the result".
And she claimed that the "overwhelming majority of people - however they voted - want us to get on with it".
The Prime Minister said she did not believe the UK would leave the EU without a trade deal, but if it did, Britain could turn the situation to its advantage.
She said: "Because we would still be able to trade with Europe. We would still be free to strike trade deals across the world. And we would have the freedom to set competitive tax rates, and embrace the policies that would attract the world's best companies and biggest investors to Britain.
"And if we were excluded from accessing the single market, we would be free to change the basis of Britain's economic model. But for the EU, it would mean new barriers to trade with one of the biggest economies in the world."
Mrs May said that the Government's plan for post-Brexit Britain would not only secure a "good deal abroad" but also "a better deal for ordinary working people at home".
She said the Brexit referendum result did not signal a "retreat from the world" but was partly due to British cultural and political history and the inflexibility of the EU in the face of David Cameron's renegotiation.
She said: "It is important to recognise this fact: June 23 was not the moment Britain chose to step back from the world, it was the moment we chose to build a truly global Britain.
"I know that this, and the other reasons Britain took such a decision, is not always well understood among our friends and allies in Europe.
"I know many fear that this might herald the beginning of a greater unravelling of the EU. But let me be clear: I do not want that to happen."
She said the UK's constitutional arrangements meant that the public expect to be able to directly hold those in power to account "and as a result supranational institutions as strong as those created by the European Union sit very uneasily in relation to our political history and way of life".
Mr Cameron's attempt to strike a better deal was a "valiant final attempt to make it work for Britain".
But she added: "The blunt truth, as we know, is that there was not enough flexibility on many important matters for a majority of British voters."
Following Mrs May's speech, Brexit Secretary David Davis will make a statement to MPs in the House of Commons on the Government's plans for leaving the EU.
Mrs May, who had backed the Remain cause in the referendum and warned about the dangers of leaving the single market, said the economy had performed better than expected.
Answering questions from British and European reporters, she said: "Just look at what has happened in terms of the economic information we have seen since the Brexit vote. What we have actually seen is that all the economic indicators have been more positive than people had predicted."
The PM said she wanted an immigration system that allowed the "brightest and best" to come to the UK but "what was clear from the June 23 vote is that as we leave the EU people want us to be able to make those decisions ourselves as to what our immigration system is for people coming in from the EU, and we will be doing just that".
Asked what would happen if Parliament rejected the deal, Mrs May said: "I am sure the British Parliament will want to deliver the views of the British people and respect the democratic decision that was taken."
Pressed on whether the main "weapon" in her armoury was the threat of turning the UK into a tax haven off the coast of mainland Europe, she said: "This is not about a 'weapon', this is not about a confrontation.
"This is about two sides coming together to negotiate and to agree a deal that is going to be in the interests of both sides."
Mrs May said the UK would always want immigration, but added: "Brexit must mean control of the number of people coming to Britain from Europe, and that is what we will deliver."
Foreign Secretary Boris Johnson said Mrs May's "powerful" speech will be "well received" by EU nations.
Speaking to reporters at Lancaster House, he went on: "Because it's negotiable, this is something that I think will be good for the UK and good for the rest of the EU as well."
Asked why the EU would give the UK a "free lunch", Mr Johnson said: "As the Prime Minister said, I think it's going to be good for both sides."
He went on: "We very strongly think this is in our mutual interest. We're not leaving Europe, we're disentangling ourselves from the treaties of the EU. We can remain powerfully committed to Europe with a new European partnership ... whilst also going forward with an identity as Global Britain.
The PM will be hoping to speak personally about her plan with German Chancellor Angela Merkel, French President Francois Hollande, European Council president Donald Tusk and European Commission president Jean-Claude Juncker in the coming days, it is understood.
Mrs May spoke to Scottish First Minister Nicola Sturgeon and Welsh First Minister Carwyn Jones ahead of her speech on Tuesday morning.
She also spoke to DUP leader Arlene Foster and Irish Taoiseach Enda Kenny on Monday, although those talks mainly focused on the political crisis in Northern Ireland.