Cardinal Sean Brady has apologised following revelations he failed to alert the authorities in 1975 about serial abuser Father Brendan Smyth.
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At a St Patrick's Day Mass in Armagh, the Catholic Primate said sorry to those who felt let down.
Cardinal Brady, who has defended his role in a meeting where two children abused by Father Brendan Smyth were asked to take a vow of silence as part of an internal church investigation, also signalled that he would reflect on his future.
'Ashamed'
"This week a painful episode from my own past has come before me. I have listened to reaction from people to my role in events 35 years ago," he said in his homily.
"I want to say to anyone who has been hurt by any failure on my part that I apologise to you with all my heart. I also apologise to all those who feel I have let them down. Looking back, I am ashamed that I have not always upheld the values that I profess and believe in."
Cardinal Brady, who has so far refused to resign despite intense pressure for him to step down, added: "Be certain that I will be reflecting carefully as we enter into Holy Week, Easter and Pentecost.
"I will use this time to pray, to reflect on the word of God and to discern the will of the Holy Spirit. I will reflect on what I have heard from those who have been hurt by abuse."
The 300-strong congregation applauded as Cardinal Brady entered the cathedral on Wednesday morning.
They applauded him again after his homily and as he left the church at the close of St Patrick's Day mass.
The head of the Catholic Church in Ireland previously said he would only step down if told to by the Pope.
On Wednesday, the Irish Primate said he looked forward to a pastoral letter from Pope Benedict on the child abuse scandals.
Pope Benedict told his weekly general audience that he would sign the pastoral letter on Friday.
The Pope expressed hopes that the efforts to address the sex abuse scandals in Ireland would help with the process of "repentance, healing and renewal".
He said the Irish church had been "severely shaken" and said he was "deeply concerned".
But Irish victims groups have already questioned the Pope's moral authority on the issue as scandals involving sex abuse in the church in his native Germany surface.
Andrew Madden, who in 1995 became the first in Ireland to go public with an abuse lawsuit against the church, dismissed the latest comments from Cardinal Brady.
"The notion of careful reflection is nonsense - he's had 35 years to reflect on what he did then," said Mr Madden.
"He's either going to go or he's not going to go and if he doesn't, the Catholic Church can't pretend to be serious in any way about the issue of child protection and about reaching out to people who have been abused."
'Dismayed'
Speaking in Washington ahead of the St Patrick's Day celebrations, Deputy First Minister Martin McGuinness said he was "dismayed" by the latest revelations to rock the Catholic Church.
"If two children were asked to sign vows of silence, then how many more were asked to do the same thing down the years?" He asked on Tuesday.
"My only conclusion to all of this is that Cardinal Brady has to consider his position".
First Minister Peter Robinson, meanwhile, said he will keep a separation between church and state.
"The important issue for the Executive is that we want to ensure that at all times children who go through the educational process are all safe."
He said Health Minister Michael McGimpsey is to report back to the Executive on calls for an inquiry into child abuse in Northern Ireland.
SDLP leader Margaret Ritchie, who is also in Washington, said Mr McGuinness should look to the controversy surrounding Sinn Fein President Gerry Adams's brother Liam, who is before the courts accused of abusing his daughter, before commenting on Cardinal Brady.
Secretary of State Shaun Woodward said Dr Brady was a "decent and caring man", adding: "I am sure he will reach the right judgment himself."
He added: "People know that Cardinal Brady is a man of great integrity, he has made his position clear."
Father Brendan Smyth was at the centre of one of the first paedophile priest scandals to rock the Catholic Church in Ireland.
A seven-month delay in extraditing him to Northern Ireland also led to the collapse of the Irish Government in November 1994.
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