A BBC investigation has revealed fresh concerns about the way the Catholic Church handles child sex abuse cases.Fr Moore's victim has spoken publicly for the first time about the abuse, and the BBC has learned the priest is still being looked after by the church.
Bishop Taylor accepted the diocese had bought Fr Moore a house to live in.
However, he said he had ordered him not to be involved in public ministry again.
The revelations come as part of a wider six-month BBC Scotland investigation into child abuse at the now-closed Fort Augustus Abbey School in the Highlands.
The investigation, which will be broadcast on Monday, heard that Fr Moore confessed to abusing more than one boy to the then Bishop of Galloway, Maurice Taylor, in 1996.
The bishop elected not to go to the police, but to send him to a treatment clinic for child sex offenders in Canada.
Bishop Taylor withheld the real reason for the priest's
departure from his parish in Prestwick, and said he was on a sabbatical
in Canada.The attached school was by this time closed, but the abbey remained open and Fr Moore joined monk Richard White there - who was also a self-confessed paedophile, later jailed for five years for child abuse.
The programme reveals that following the abbey's closure in 1998, Bishop Taylor tried to get Fr Moore a job saying mass for nuns at a nursing home. They refused after hearing about his admissions about child abuse.
The BBC spoke to one of Fr Moore's victims, who was abused by the priest when he was a young altar boy in the late 1970s.
Tony, not his real name, is speaking publicly for the first time.
He said: "(The priest) was somebody you looked up to, you trusted, you believed in. And you would not think they would do any wrong.
"One of my memories was when it was just the two of us on the beach when we were getting dressed - he (took) an interest in drying me. He (got) quite aroused by this."
Fr Moore went on to sexually assault the 11-year-old.
'Removed him'
Tony told the BBC he was offered no support by the church after Fr Moore's confession, and even bumped into his abuser in a street in Ayrshire.
Fr Gerard Magee, a priest based in Kilwinning, criticised the church's handling of the case.
He said: "Bishops have bent over backwards in the past to protect the good name of the church, such as it is, and to protect their own credibility, such as that is, but when we're talking about child abuse you know you have to ask the question what's creditable about child abuse? Nothing."
He added: "It makes me ashamed. Not only ashamed of what happened to the children, but what was worse was the cover-up by the hierarchy in the church."
The BBC documentary also reveals that Fr Moore is living in a house which was purchased by the church.
In a statement, Bishop Taylor said he did alert the authorities to Fr Moore's child abuse admission.
However, this did not happen until eight months after Fr Moore's confession.
He said: "In April 1996, Fr Moore told me of actions which had occurred years previously. On his admission of these to me I removed him from the pastoral ministry.
I sought advice on how to handle Fr Moore's admission from social workers, experts in child protection and legal figures.
"The initial advice I was given was that since no allegations had been made against Fr Moore but that he had made personal admission to me, I didn't need to inform the authorities."
Bishop Taylor denied misleading the parish about Fr Moore's whereabouts.
He said: "I have no memory of describing Fr Moore's leave of absence as a sabbatical; in The Galloway Newsletter he was simply described as having leave of absence. At no point did I have the intention to deceive the parishioners."
The bishop said he arranged a meeting with the procurator fiscal in Kilmarnock in November 1996.
"He told me that he would be informing the police of what I had told him," Bishop Taylor said.
"The Crown Office informed us in 1999 that they had decided not to proceed with any action but the case remained open."
'Done differently'
He added: "A house was purchased... for his use, I cannot remember the price but we were spending around £40,000 for such properties at that time. The house remains the property of the diocese.
"I was never made aware of who the 'alleged victims' were… I could not approach people whose identity was unknown to me.
"Things would have been done differently in some aspects if we had had the Guidelines for Safeguarding that the Church now operates with."
Bishop Taylor said he had not tried to appoint Fr Moore as chaplain at a nursing home, merely to say Mass there.
He added that the suggestion was rejected by the Mother Superior, and it was a decision he "wholeheartedly accepted".
Fr Moore told the BBC: "I regret if any person believes that past activities of mine were abusive: that was certainly never my intention, or my understanding of the matters discussed with Bishop Taylor.
"Those matters were fully investigated by the police, and the Procurator Fiscal confirmed in 1999 that no proceedings should be taken in relation to them."
For the past six months, the BBC has been investigating allegations of serious physical and sexual abuse at Fort Augustus Abbey School over a 30-year period.
The programme contains testimony from victims, speaking openly for the first time about about their experiences at the hands of Benedictine monks.
You can watch BBC Scotland Investigates: Sins of Our Fathers on on Monday 29 July on BBC1 Scotland at 21:00, and for a week afterwards on the BBC iPlayer