Businessman Gilbert Nulty, 65, was jailed for eight years in 2000 for sexually abusing three young girls while he was a part-time police officer.
His conviction was thrown out three years later after appeal judges ruled his right to a fair trial was infringed by the admission of hearsay evidence from one of his alleged victims.
However, one of his victims launched a civil claim for damages and has now been awarded £70,000 after a 10-year battle.
Judge Lord Kinclaven said he was satisfied Nulty had abused the woman over an eight-year period when she was a child.
In a written ruling, he said: "Having seen and heard the witnesses, I was so satisfied in relation to events while the pursuer was a child and the consequences thereof.
"In relation to the defender, having seen and heard the whole evidence, I reject the defender's evidence as being neither credible nor reliable.
"There was a certain arrogance about the defender when he gave his evidence. I found his answers unconvincing.
"In sharp contrast, I accepted and preferred the evidence of pursuer in relation to the defender's offending. The pursuer's account has been consistent over the years."
Nulty, of Beith, Ayrshire, was alleged to have committed the offences between 1977 and 1987. His role as a special constable ended after 14 years' service in 1989.
His victim, now aged 45 had originally sued Nulty for £250,000.