The Government must then hold a full debate on the issue in the Oireachtas within four months – and if there is a vote in favour, it must then include a timeline for a referendum.
But a government spokes-man declined to be drawn on when this might be.
"The Government will, as promised, take under consideration the recommendation of the Constitutional Convention in due course," he said.
The three strongest supporters of gay marriage in the cabinet were quick to hail the result. Tanaiste Eamon Gilmore, who has described it as the civil rights issue of our generation, said he was always confident that people would see the wisdom of the argument.
"It's not the role of the State to pass judgment on who a person falls in love with, or who they want to spend their life with," he said.
Children's Minister Frances Fitzgerald said the 79pc majority vote in favour of a gay marriage referendum at the Constitutional Convention had sent a powerful message to the Government.
"If you were to take a temperature in relation to this issue, I think the Government would have to be very aware of the strength of feeling in favour of it," she said.
And Justice Minister Alan Shatter, who has also declared his support for gay marriage, promised to bring forward legislation to give gay and lesbian parents the same rights as heterosexual parents.
He said his Family Relationships and Children's Bill would address issues relating to the "parentage, guardianship and upbringing of children".
The support for a referendum on gay marriage was also welcomed as a "historic step" by a host of campaigners, including the Gay and Lesbian Network, the Irish Council for Civil Liberties and lobby group Marriage Equality.
ICCL director Mark Kelly said the vote sent a clear message to Government that the time was right to ask the Irish people to remove one of the "last bastions of official discrimination in Ireland".
The trio has called on Mr Shatter to press ahead with legislation to regulate the relationship between parents and children in same-sex families.
However, the Catholic hierarchy, which made representations at the convention to maintain the constitutional status quo, described the poll as "disappointing".
It said that only the people of Ireland could amend the constitution.
"The Catholic Church will continue to promote and seek protection for the uniqueness of marriage between a woman and a man, the nature of which best serves children and our society," said Bishop Leo O'Reilly, who addressed the convention.
The Iona Institute, the pro-marriage think-tank, said the result in a referendum "outside of the artificial atmosphere of the convention" would be very different.
And Senator Ronan Mullen, who unsuccessfully sought to make amendments to the draft ballot to expressly protect religious freedom, described the convention as a "flawed process".