Gilroy, 49, was found guilty of murdering Ms Pilley, 38, in March, almost two years after she set off on her usual journey to work in the city centre but never arrived.
Lord Bracadale gave the killer, said by prosecutors to be “a deceitful and controlling” individual, a life sentence and told him he must spend a minimum of 18 years behind bars before he could be considered for release.
A Scottish Courts spokeswoman said Gilroy was now appealing against his conviction, with a report expected by August 15 for a “sifting” judge to decide if the appeal has merit.
The jury in Gilroy’s trial heard that he was driven by jealousy, maintained a front of normality and embarked on a series of acts to cover up his crime.
He took Ms Pilley’s body to a secret grave, believed to be in remote Argyll.
Married Gilroy, who has protested his innocence since the day Ms Pilley vanished, was convicted of murder after a four-week trial.
Lord Bracadale told Gilroy he hoped he would one day say where he abandoned the victim’s body.
Gilroy also made legal history when he became the first convicted killer to have his sentencing filmed for British TV.