Lawyers for the 49-year-old presented an intimation to appeal yesterday morning at the Appeal Court in Edinburgh to start their bid to free Gilroy.
A transcription of the summing-up presented by judge Lord Bracadale to the jury before they started their deliberations will now be prepared for Gilroy’s legal team to scrutinise. It is understood the team will look for any flaws in the judge’s charge, which could provide grounds for the appeal.
Gilroy’s solicitors now have eight weeks to lodge the grounds of appeal on which their client will rely, although this deadline can be extended under particular conditions.
Gilroy is appealing against his conviction for murdering Ms Pilley and disposing of her body, but has not launched an appeal against the minimum 18-year sentence he was given at the High Court in Edinburgh last Wednesday.
Peter Durrand, the former husband of Ms Pilley, said he hoped the appeal was not granted, to spare her family further anguish.
The 49-year-old said: “It’s not unsurprising that this appeal has been made. It’s hard to see what his grounds of appeal will be.
“I cannot see what faults his lawyers will be able to find in the case, but I suppose that’s what they’re paid to notice.
“Obviously, we are hoping that the appeal is not granted for the family’s sake.”
Mr Durrand, who was with Ms Pilley for seven years and now lives in Inverkeithing, said previously he believed the convicted killer would become “another Luke Mitchell”, who maintains his innocence and files an “endless” barrage of appeals.
He said: “I still think that’s the case. These things do seem to go on and on. They say it’s justice but it allows people to appeal again and again. Maybe Gilroy is paying for this himself but these appeals often come out of the public purse.”
Wardlaw Stephenson Allan, the firm representing Gilroy, was contacted for comment yesterday but did not respond.
An appeal court spokesman confirmed an intimation to appeal had been received on behalf of Gilroy.
Gilroy, a married man, was found guilty last month of murdering Ms Pilley, his 38-year-old former lover who worked with him at Infrastructure Managers Ltd in the city centre.
Ms Pilley sparked a major missing person hunt when she disappeared on May 4 2010. Prosecutors believe that Gilroy killed her in the basement of the building where they worked in Thistle Street and buried her the next day more than 100 miles away in the forests of Argyll.
Passing sentence, Lord Bracadale said Gilroy was the only person who knew the where- abouts of Ms Pilley’s body, which has never been found.
Lord Bracadale also spoke of the “quite chilling calmness” with which Gilroy set about disposing of her remains.
After Gilroy was sentenced last Wednesday, the Gilroy family released a statement which read: “As we said on March 15, our family is devastated by the verdict. We continue to believe that David is not guilty of the charges of which he was convicted by a majority decision.”