Epstein had pleaded guilty in 2008 to procuring for prostitution a girl under the age of 18 and served 13 months in prison before being released on probation.
Epstein, 66, died by apparent suicide last week while he was being held at the Metropolitan Correctional Center in Manhattan as he awaited trial for sex trafficking and conspiracy charges.
Buckingham Palace doesn't usually comment on scandalous accusations related to royals' personal lives - their typical approach is to raise the drawbridge and simply say "no comment".
"His Royal Highness deplores the exploitation of any human being and the suggestion he would condone, participate in or encourage any such behavior is abhorrent", Buckingham Palace wrote.
It shows Andrew peering out from a large door of the mansion waving goodbye to a woman after Epstein leaves to get in a chauffeur-driven auto.
The UK's Prince Andrew has said he was "appalled" by allegations of sexual abuse surrounding Jeffrey Epstein after a video was released purporting to show him at the home of the convicted paedophile in 2010.
The video was reportedly shot on 6 December 2010, around the time the Duke of York was photographed with disgraced billionaire Epstein in New York's Central Park.
The well-connected American financier, whose circle of friends and acquaintances included a long list of politicians, socialites, and celebrities including Donald Trump before he was president and former US President Bill Clinton.
He quit his role as United Kingdom trade envoy in 2011 after the fallout from the Central Park photos.
Andrew's connections to Epstein came under fresh scrutiny following newly unsealed legal documents on August 9.
However, the Palace said the allegations are "false and without any foundation", stating that "any suggestion of impropriety with under-age minors" by the duke was "categorically untrue".
The FBI and the Justice Department's inspector general are continuing to investigate Epstein's death.
Virginia Roberts claimed she was forced to have sex with the prince on three occasions - in London, New York and on a private Caribbean island owned by Epstein - between 1999 and 2002, when she was under-age according to U.S. law. But the palace has repeatedly issued statements denying any wrongdoing by Andrew.