The Queen has held a meeting with Irish President Mary McAleese for the first time on the island of Ireland.
The Queen and President McAleese met at Hillsborough Castle
The meeting at Hillsborough Castle intensified speculation about the timing of any future visit by the monarch to the Irish Republic.
In June, President McAleese indicated she would welcome a visit to the Republic of Ireland by the Queen.
On Thursday, Mrs McAleese said: "It was a very special day for Anglo-Irish relationships."
A decision on when the Queen should visit the Republic
was not down to her, she said.
"I'm long on record as saying that I would welcome it very warmly.
"But it's a matter for the two governments.
"Earlier this year, after the Co-Operation Ireland dinner, it was said then that the two governments were in agreement that the visit should take place."
The timing of that visit would depend on the "final consummation of the peace process in Northern Ireland", she said.
"I think things are developing. Things are going in the right
direction, so the day gets nearer."
Earlier on Thursday, the Queen visited the Belfast Central Mission which works with the elderly and young people.
The Queen and Duke of Edinburgh arrived on Wednesday and stayed overnight at Hillsborough Castle.
The couple were on a one-day visit and attended separate engagements in County Down.
The Queen and Duke had lunch with NI secretary Peter Hain, the NI Lord Lieutenants, NIO Permanent Under Secretary Jonathan Phillips and Nigel Hamilton, head of the NI Civil Service.
The Queen met a mother and toddler group in Hillsborough
In Hillsborough, she officially opened buildings housing Downshire Primary School and the new village community centre.
During the event, she was entertained by a choir, visited classrooms and also met a mother and toddler group.
The Queen also unveiled a plaque and planted a tree.
In Belfast, Prince Philip visited the headquarters of the Duke of Edinburgh award scheme in the south of the city.
The monarch and President McAleese have met on three previous occasions.
In 1998 they attended the unveiling of a peace tower on the site of a World War 1 battle in Belgium and the following year the president attended a lunch at Buckingham Palace.
In June this year, both were at a Cooperation Ireland dinner in London.
At that time, President McAleese said the Irish and British governments were agreed a visit should take place and the timing "is for decision by them in the light of the successful development of the political process in Northern Ireland over time".
The Queen visited Belfast Central Mission which works with the elderly
The Queen last visited Northern Ireland in February 2003, when she unveiled a plaque to commemorate the opening of the Laganside Courts complex.
She and the duke also visited in July 2002 as part of the celebrations marking her Golden Jubilee.
Since her coronation, the Queen has visited Northern Ireland on 13 occasions, on 11 of which she was accompanied by the Duke of Edinburgh.
The duke has visited Northern Ireland a further 24 times.