The President, Mrs McAleese last night said it is "time to close the door on the tradition of armed struggle".
In a speech welcomed by Mr Martin McGuinness of Sinn Féin, Mrs McAleese said it was time to make "a decent start to the shared future that is the unarguable entitlement of the next generation".
Speaking at the end of a politically sensitive visit to Belfast, Mrs McAleese called for a renewed effort to advance the peace process.
Appealing implicitly to the IRA and its supporters, she said: "It is time to close the door on the tradition of armed struggle and to bring a dignified and principled end to the debate started by Daniel O'Connell. . . . It is time to make a hope-filled, humanly decent start to the shared future that is the unarguable entitlement of the next generation".
Speaking in Cork last night before a Sinn Féin fundraiser, Mr McGuinness said everyone had "a duty to listen very intently to what the President says.
"There is no doubt whatsoever that whenever she speaks, people, particularly from the nationalist-republican tradition, place a lot of store in what she has to say."
Asked if the IRA would heed what she said, Mr McGuinness said: "I hope that everyone would heed what the President has to say, and from our perspective in Sinn Féin, we are not going to shirk the difficult questions and challenges that lie ahead."