Cardinal Seán Brady has publicly apologised to Brendan Boland, a survivor of paedophile priest Fr Brendan Smyth.
Speaking at Lough Derg following a penitential pilgrimage in advance of next month's Eucharistic Congress, Dr Brady also said he had no intention of stepping aside, despite continuing calls for his resignation.
He said there had also been "many many calls from people who want me to stay on."
But he said he hoped a coadjutor - with succession rights - would be appointed to his archdiocese as soon as possible.
Dr Brady said he now realised that the parents of children who were being abused by Brendan Smyth should have been informed about the allegations of abuse being made against him.
"Definitely the parents should have been informed. That's quite clear,” he said.
Dr Brady said he apologised without hesitation to Brendan Boland, and to all survivors of abuse.
He said he would also like to personally apologise to Mr Boland, and hoped to do so in the future.
He said he intended to remain on as primate "until I'm 75, or unless the Holy See indicated it didn't want me to stay".
The cardinal said there was absolutely no indication from the Holy See that it wanted him to resign.
Cardinal Brady's apology to Brendan Boland follows comments by the Bishop of Ardagh and Clonmacnoise about the issue on Sunday.
Speaking yesterday to RTÉ's This Week programme, Colm O'Reilly said he could not understand why the Cardinal had not publicly apologised to Mr Boland.
He said that the Catholic Church "should be doing everything we can" to meet the requests of abuse survivors.
Bishop O'Reilly told the programme that it would not take much and would not take long for Brendan Boland's request for a public apology to be met.