Mr Kenny was driven to Phoenix Park, arriving for his engagement with the President shortly after 2.30pm.
He was taken to the State Reception Room, where Mrs McAleese signed the Warrant of Appointment of Taoiseach and presented him with the seals of office.
The new Taoiseach spent a half an hour with his family and the President and her husband, Dr Martin McAleese, in the Aras.
As he left he was asked how it felt to be Taoiseach.
“Well, it feels very different, actually, but I hope I’m the same,” Mr Kenny said.
And when asked if it was good day for Ireland, he added: “I hope so, and I hope I can prove it.”
The Dáil has now been adjourned until 4.40pm, at which time we will learn the make up of Mr Kenny's new Cabinet.
Fine Gael's Michael Noonan is widely expected to get the coveted finance portfolio, while Labour is tipped for the Attorney General post, giving the smaller party potentially six seats at the Cabinet table.
Other Fine Gael TDs tipped for a ministry include Programme for Government negotiators Phil Hogan and Alan Shatter, as well as Richard Bruton, Leo Varadkar, Sean Barrett, James Reilly, Simon Coveney and Jimmy Deenihan.
Labour hopefuls include former leaders Pat Rabbitte and Ruairí Quinn, Roisín Shortall and Brendan Howlin, with the party's finance spokesman, Joan Burton, widely expected to get the new finance brief with responsibility for expenditure and public sector reform.
Current Labour leader Eamon Gilmore is expected to take on a new foreign affairs and trade brief, as well as becoming Tánaiste.
Mr Kenny will face his first official engagement on Friday when he travels to Brussels for an extraordinary European Council meeting, where he will meet other heads of state.
In Government, the two parties have pledged to seek better repayment terms for last December's €85bn EU/IMF rescue package.
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