The head of the Catholic Church in Ireland said the priest had agreed to take leave from his ministry.
"The policy of the Archdiocese of Armagh is that in all matters relating to child safeguarding, the safety and welfare of the child must be our paramount concern," he said.
He said he had asked the priest concerned to take a period of leave from his ministry, which he agreed to.
"This is to allow the civil authorities, who have been informed, to investigate this matter," he said, adding that the priest continued to enjoy the right to the presumption of innocence while matters were investigated.
Cardinal Brady also used the opportunity to invite anyone who might have been abused by a priest or member of a religious order to come forward to the diocese and/or to contact the civil authorities.
Meanwhile, the Archbishop of Dublin Diarmuid Martin has appealed to Catholics considering leaving the Church because of the way it handled reports of clerical sex abuse to join in taking responsibility for it from within.
"If we really understand how we all belong to the one body then we cannot feel that the answer to renewal in the Church can come about by leaving the Church, or by leaving it to others," said Dr Martin when he addressed a congregation in Dublin’s Pro-Cathedral.
"These have not been easy days for me personally. But, with the many believers who wish to journey together on the path of renewal, I know that path will inevitably be a way of the cross," he said.
Meanwhile, ex-bishop of Cloyne Dr John Magee has thanked people of the diocese for their prayers and support over the last 23 years and apologised for his failings.
In a statement read out at masses in Cloyne, Co Cork, yesterday, Dr Magee said he was privileged to have served as head of the diocese. The former papal secretary stepped aside from his duties a year ago when it was revealed that he did not follow proper child protection guidelines.
His resignation was accepted by Pope Benedict last week.
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