Theresa May has said she is "honoured and humbled" to become new leader of the Conservatives.
It was announced today by David Cameron that she will take up the post on Wednesday after he has taken his last Prime Minister's Questions and offered his resignation to the Queen.
She will become the second female British prime minister, after Margaret Thatcher.
In a statement outside the Commons, Britain's next Prime Minister praised David Cameron for his stewardship of the party and country.
Mrs May also hailed Andrea Leadsom, whose shock decision to pull out of the Tory leadership race paved the way for her coronation, for the "dignity" she had shown.
Flanked by dozens of applauding MPs from all sides of the party, Mrs May said she would offer "strong, proven leadership to steer us through what will be difficult and uncertain economic and political times".
She said that she would "negotiate the best deal for Britain in leaving the EU and forge a new role for ourselves in the world".
In a message designed to reassure Eurosceptic Tories that her support for Remain during the referendum does not mean she will seek to go back on the vote to leave the EU, Mrs May said: "Brexit means Brexit and we're going to make a success of it."
And she added: "We need to unite our country and ... we need a strong, new positive vision for the future of our country - a vision of a country that works not for the privileged few, but that works for every one of us.
"Because we are going to give people more control over their lives and that's how together we will build a better Britain."
Mrs May's husband Philip was at the victory speech outside St Stephen's entrance and gave his wife a kiss on the cheek as she was cheered by supporters.
Chris Grayling rejected calls for a snap election, stressing that Mrs May was a senior member of the Tory team that won the general election last year.
The Cabinet minister told the Press Association: "It's 15 months since the Conservative Party got a mandate with her as one of its key leading members - I think the last thing this country needs right now is a general election."
He paid tribute to Mrs Leadsom's "noble" decision.
"I think today's events have been a little unexpected but I would pay tribute to Andrea Leadsom for putting the country's interests and the need for strong leadership at a difficult time ahead of her own interests.
"I think that's absolutely right for her to have done that, she's been very noble in putting her interests second - it shows that she's a true public servant," he said.
The Commons Leader said the rapturous reception received by Mrs May at the 1922 Committee and outside Parliament showed that she commands the support of the vast majority of Tory MPs, adding: "The parliamentary party will now unite behind her ".
Mr Grayling, who has been tipped for a Cabinet job after playing a leading role in Mrs May's campaign, said he had not asked for a job.
"What she decides to do about her Cabinet is entirely up to her - I've not asked for a job, nor would I expect to be offered one as part of the campaign," he said.
Tory MP and close ally of Mrs May, Damian Green, insisted that the new prime minister did not need to call a general election.
When asked if Mrs May had a mandate, Mr Green told BBC Radio Four: "I think she does because she was a very senior member of a Government that was elected just over a year ago.
"We don't elect presidents in this country. We elect a parliament, we elect MPs for individual seats.
"And the question the Queen asks formally, constitutionally, is 'can you command a majority in the House of Commons?' And Theresa can command a majority, so there is no need for an election."