It is claimed that the late Robert Waddington groomed an 11-year-old choirboy for five years during the 1980s.
This choirboy, Eli Ward, who is now 40 years old, waived his right to anonymity in the press when he made the claims.
The Archbishop of York, John Sentamu, set up the inquiry last year to find out what information was given to Church officials at the time of the alleged abuse, and whether any improper conduct was covered up.
In her findings, Judge Sally Cahill QC, found that church officials including a former Archbishop of York, didn't properly address the issue.
She said: "Our conclusion, having heard his [Lord Hope's] evidence is that his concern for the welfare of Robert Waddington seems to have been paramount in his response to these allegations."
Responding to the findings, the current Archbishop of York, Dr John Sentamu, said: "I have already been in contact with those who gave evidence to the Inquiry regarding their alleged abuse by Robert Waddington.
"As I have said to them, I am deeply ashamed that the Church was not vigilant enough to ensure that these things did not happen, failing both to watch and to act, where children were at serious risk.
"Any act of abuse committed by someone in a position of authority in the Church is a matter for shame and requires deep repentance.
"We are called as individuals and corporately to a higher standard and to show God's love and care as revealed in Jesus Christ.
"Those who trusted us in this only to be grievously wounded deserve not only our wholehearted apology but also the assurance we will keep a watchful eagle's eye and act swiftly."
The Bishop of Manchester, Rt Revd David Walker, has described the details of the report as chilling, speaking to Premier, he said: "It makes my blood run cold, I mean how anybody could behave like that and the ongoing impact on people who suffer such abuse. This is not something that they can recover from, they take that harm, that hurt; that pain with them for the rest of their lives, it's a chilling story."
Lord Hope has said he is "disappointed" that the report has raised concerns about how the cases were handled and denied that there was a cover-up.
Watch an interview with Dr John Sentamu here:
Listen - Bishop of Manchester, Rt Revd David Walker: