It comes after several Sunday newspapers published comments by Virginia Roberts and identified her as the woman who reportedly named Andrew in US court documents related to a convicted paedophile.
The palace initially said “any suggestion of impropriety with under-age minors” by the Duke was “categorically untrue”.
In a second strongly worded statement last night, a Buckingham Palace spokesman said: “It is emphatically denied that the Duke of York had any form of sexual contact or relationship with Virginia Roberts.
“The allegations made are false and without any foundation.”
Ms Roberts, a US citizen, waived her anonymity in an interview with the Mail on Sunday in 2011, in which she claimed she had been sexually exploited by American billionaire and convicted sex offender Jeffrey Epstein — a former friend of the Duke.
Andrew was photographed with Epstein in New York two years after the American’s release from prison in 2009, and their friendship was a source of controversy.
In the US court papers, the woman, reported to be Ms Roberts, claims that between 1999 and 2002 she “was forced to have sexual relations with this prince when she was a minor” in London, New York and on a private Caribbean island owned by Epstein.
The accusation is reportedly contained in a motion filed in a Florida court this week which is part of a lawsuit over how federal prosecutors handled the case of Epstein, who was jailed for 18 months in 2008 for soliciting a minor for prostitution.
The Mail on Sunday and the Sunday Mirror have since published details of interviews given by Ms Roberts in recent years, in which she discusses three alleged encounters with the Duke.
Andrew is believed to have been on a ski trip in Verbier, Switzerland, as the latest reports of the controversy broke. A royal source confirmed the Duke had been on a private holiday abroad.
It is understood that despite the controversy, he is planning to attend the World Economic Forum annual meeting in Davos, Switzerland, later this month.
In the court papers, Epstein is accused of repeatedly abusing the woman — now said to be in her 30s — on his private jet and luxury homes in New York, New Mexico, Florida and the US Virgin Islands.
The court document says: “Epstein also sexually trafficked the then-minor Jane Doe (a name used in US legal proceedings for people with anonymity), making her available for sex to politically connected and financially powerful people.
“Epstein’s purposes in ‘lending’ Jane Doe (along with other young girls) to such powerful people were to ingratiate himself with them for business, personal, political and financial gain, as well as to obtain potential blackmail information.”
It adds: “For instance, one such powerful individual Epstein forced Jane Doe to have sexual relations with was a member of the British royal family, Prince Andrew (aka Duke of York).”
Harvard law professor Alan Dershowitz — who is also named in the court papers — said the claims against him were part of a pattern of “made-up stories” against prominent people.
Prof Dershowitz, who represented Epstein in the 2008 case, said he would not rest until he had shown that the claims regarding his conduct were false and he advised the Duke to do everything in his power to clear his name.
“My only feeling is, if she’s lied about me, which I know to an absolute certainty she has, she should not be believed about anyone else,” he told the BBC Radio 4 Today programme.
In a statement released through her lawyers to several newspapers, the woman hit back, saying: “These types of aggressive attacks on me are exactly the reason why sexual abuse victims typically remain silent and the reason why I did for a long time. That trend should change.
“I’m not going to be bullied back into silence.”