Bishop James Moriarty admitted in December that he had not challenged the Dublin archdiocese over concealing child abuse complaints from police.
He served as an auxiliary bishop in the Dublin archdiocese between 1991 and 2002.
The joint statement issued by bishops will go out to all parishes in England and Wales - it said those who had carried out the abuse brought "deep shame to the whole Church".
"Catholics are members of a single universal body. These terrible crimes, and the inadequate response by some church leaders, grieve us all," it said.
The statement went on: "We express our heartfelt apology and deep sorrow to those who have suffered abuse, those who have felt ignored, disbelieved or betrayed.
"We ask their pardon, and the pardon of God for these terrible deeds done in our midst. There can be no excuses."
And it said the Church would work with safeguarding commissions within its dioceses to ensure relevant steps were taken to protect against any further abuse and atone for those who were already victims.
The statement was presented by Vincent Nichols, the Archbishop of Westminster, who said the words were "very heartfelt" and "unambiguous".
Archbishop Nichols told BBC News: "We say there are no excuses.
"We simply apologise profoundly for the hurt and the lasting damage that is done through childhood abuse, and for the inadequate ways in which in the past these things have been handled."
In recent months there has been a wave of child abuse reports across the globe against the Catholic Church.
In recent months victims have come forward in the Irish Republic, Austria, Switzerland, Italy, the Netherlands, Germany and the US.
And critics have accused the Church of failing to deal adequately with the problem.
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