It is "time to close the door on the tradition of armed struggle", the President said last night.
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 McGuinnes 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."
It was Mrs McAleese's first appearance in Belfast since her remarks about intolerance and bigotry on the 60th anniversary of the liberation of Auschwitz offended many Protestants.
She used the O'Connell Lecture at St Malachy's College in north Belfast to make a powerful appeal for renewed efforts to resolve issues blocking political progress.
The Belfast Agreement "was built on a shared hope and a capacity for mutual forgiveness unequalled in any conflict situation in modern times with the exception of the creation of the European Union out of the appalling suffering of World War II," she said.
It was time to "reflect on the opportunity we have so nearly in our grasp which so many have invested in and so much of our future depends upon". It was also the moment for "conscientiously seeing things through, for making good on promises given and accepted in good faith".
The speech concluded a successful series of engagements in Belfast. Despite the postponement of a visit to a school in the loyalist Shankill following an outcry from unionist politicians and community leaders over her earlier comments, Mrs McAleese's visit was endorsed by a senior UDA commander.
Mr Jackie McDonald, a senior UDA figure in south Belfast, commended the President and her husband, Dr Martin McAleese, saying they had offered the hand of friendship and Protestants would not forget that. "I have no doubt that, in time, she will be welcome on the Shankill Road," Mr McDonald said.
Mrs McAleese did not visit a primary school in the Shankill area as had originally been planned. However, sources close to the President insisted the visit was "deferred" rather than cancelled and it is understood a visit is being rescheduled later before the end of the school year.