The President, Mrs McAleese, on her first visit to Britain since her election, spoke repeatedly of the Irish and British people standing on "the threshold of a new and very healthy phase" in their relationships.
"You show how easy it is for two great cultures to happily live side by side, deferring to each other and respecting each other," she told an audience in the Leeds Irish Centre. "It is possible to live with difference, and to live generously in spirit, to live happily and contentedly with two cultures."
She spoke of a "breakthrough" in the peace process and said on the eve of the new millennium everyone was hoping for and looking forward to an era of "genuine and real friendship between these two islands and all their peoples".
She said a space had been made in Britain for Irish people "to enter the warp and weft of British life" and on the other side Irish people and their different culture had brought a "richness and diversity" to Britain. This was particularly so in Leeds, where the Irish community had been founded by Irish weavers who had come to Yorkshire in the early years of the 19th century following the collapse of their industry.
Mrs McAleese repeated her hope that there would be an exchange of state visits between the British Queen and herself. Such an exchange would "say something about the state of maturity we have achieved in our sets of relationships".
Earlier, the President addressed the Catholic Independent Schools Conference at Ampleforth College in north Yorkshire, honouring an invitation she had received before her election.
Mrs McAleese said the Christian gospel called for "forging partnerships with those from whom we are estranged by history, tradition, class, religion, culture, upbringing and politics".
She praised those who disturbed the "God is exclusively on our side" version of history in Northern Ireland. "There have been a small number of noble attempts in very recent years to acknowledge the dark side of the past, to offer sorrow for it, to draw a line in the sand which helps us to move on.
"Such gestures, particularly from those in positions of leadership, can have surprisingly profound significance in a world where centuries-old subterranean hurts still fester, infecting each new generation, handing on the baton of bitterness from one century to the next. Little acts of generosity have the capacity to soften hearts grown cold with cynicism."
In the North, there were "two cultures, two identities, inhabiting the same spot but travelling in their hearts towards very different destinations," she said.
"Today's and tomorrow's work is to change the destination to one on which all are focused without emasculating either culture or obliterating either identity. The creation of a working partnership out of difference will demand that we make a new space in our thinking to accommodate the `otherness' of the other, that we look at their `otherness' in ways that are much more generous than in the past."
After her address, the President chatted to a dozen Irish boys at Ampleforth - one of England's leading Catholic public schools - including the three sons of fashion designer Paul Costelloe.