Published Thursday, 17 December 2009
Bishop Murray will leave day-to-day diocesan duties in Limerick but retain his title.
He apologised in person to mass-goers in St John's Cathedral in Limerick.
"I humbly apologise once again to all who were abused as little children," he told the congregation, which included priests of the diocese.
"To all survivors of abuse, I repeat that my primary concern is to assist in every way that I can, on their journey towards finding closure and serenity."
Bishop Murray's resignation comes weeks after retired Bishop Dermot O'Mahony quit as patron of The Irish Pilgrimage Trust.
They are the first clergymen to stand down following the Murphy report.
Bishop Murray offered his resignation to Vatican officials on December 7, after pressure mounted from within the hierarchy, including calls from Cardinal Brady for him to do the right thing.
His mishandling of child abuse cases against priests in Dublin was in one case inexcusable, the inquiry found.
"I have heard the views of many survivors, especially in the days following the publication of the Murphy Report," he told the congregation on Thursday.
"Some expressed the wish that I should resign; others asked me not to do so.
"I know full well that my resignation cannot undo the pain that survivors of abuse have suffered in the past and continue to suffer each day."
Bishop Murray added: "I asked the Holy Father to allow me to resign and to appoint a new bishop to the Diocese because I believe that my presence will create difficulties for some of the survivors who must have first place in our thoughts and prayers."
Commenting on the resignation, Cardinal Sean Brady apologised again "to all who were abused as children by priests, who were betrayed and who feel outraged by the failure of Church leadership in responding to their abuse."
Archbishop of Dublin Diarmuid Martin stopped short of calling for further resignations but said: "Responsibility must be taken by all who hold a position of authority and collective responsibility."
But support group One In Four said one resignation did not signal a major change in policy in the church.