The pair met in Kelvingrove Park in the city's West End, where the cleric often went to use the services of male prostitutes.
Gallagher, 30, found out that his client was a priest and began extorting money from him, threatening to go to the press if he didn't pay up.
At Glasgow Sheriff Court yesterday, Gallagher, of the Talbot Association, Paisley Road, Glasgow, admitted menacing the priest, threatening to expose him as homosexual and demanding money from him on various dates between 1 January, 2005, and 16 September, 2007.
Prosecuting, Judith Hutchinson told the court that the victim had been a priest for almost 33 years when he met his blackmailer in Kelvingrove Park about five years ago.
He paid Gallagher to perform sex acts on two occasions, but also regularly spoke to him and gave him money when he was at the park using other prostitutes because he "felt sorry for him".
Miss Hutchinson said: "Around 2005 the accused's attitude towards the complainer changed. It became apparent that the accused was aware that he was a priest; he started to call him 'Father'.
"The complainer became worried that his activities would be uncovered and a short time after that the accused began to demand money from him.
"The accused repeatedly threatened to expose the complainer as a practising homosexual and told him that if he didn't give him money he would go to the press."
The cleric carried on giving Gallagher cash when he saw him at the park, until July last year, when Gallagher followed him and found out where he lived. He showed up at the chapel house the next day and demanded money.
Over the next few months Gallagher appeared at the manse at least once a week to receive payment, and even forced his way into the house when he was refused.
On 2 September last year, Gallagher turned up at 7:30am and barged his way into the house. He threatened the priest and forced him to write a cheque for £2,220.
He also demanded that his victim write a letter stating that he was gay and used male prostitutes.
Later the same month, local police received information that a priest in the area was being blackmailed. Officers visited the priest, who told them what had been going on.
Gallagher was later traced and arrested at his home address.
Miss Hutchinson added: "The complainer resigned as a result of this case coming to light and he now admits to being a practising homosexual who uses the services of male prostitutes."
Sheriff Norman Ritchie deferred sentence until next month for reports and Gallagher was released on bail.
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