The court papers claim that the Duke carried out "acts of sexual abuse" against the girl, who was 17 at the time, which is considered to be a minor in Florida. The girl's age suggests the allegations date to 2001.
The Duke is not a party to the court action and has not been given the opportunity to rebut the allegations through the court.
Buckingham Palace normally has a policy of refusing to comment on ongoing court cases, but the nature of the allegations, and the fact that they had been reported in the mainstream media, forced the Palace into having to issue a formal denial that the Queen's son was a sex abuser.
A Buckingham Palace spokesman said: "This relates to long-running and ongoing civil proceedings in the US to which the Duke of York is not a party.
"As such we would not comment on the detail. However, for the avoidance of doubt, any suggestion of impropriety with underage minors is categorically untrue."
The allegations were made in the latest round of a six-year-old lawsuit brought by women who say they were exploited by Epstein and who claim they should have been consulted before he was allowed to enter into a plea bargain agreement with prosecutors.
Two women, named only as Jane Doe 3 and Jane Doe 4, have asked to be included in the action brought by two other unnamed women.
Jane Doe 3 claims in the papers, lodged in the US District Court Southern District of Florida, that in 1999 she was approached by Ghislaine Maxwell, the daughter of the late newspaper owner Robert Maxwell, who would "procure" under-age girls for sex with Epstein. Ms Maxwell has always strenously denied any involvement.
Jane Doe 3, who was 15 at the time, claims she was "converted into a sex slave" and was kept as such until 2002, when she "managed to escape".
The court document states: "Epstein sexually trafficked the then-minor Jane Doe, 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 blackmail information...
"One such powerful individual Epstein forced Jane Doe 3 to have sexual relations with was a member of the British Royal Family, Prince Andrew (a/k/a Duke of York).
"Jane Doe 3 was forced to have sexual relations with this Prince when she was a minor in three separate geographical locations: in London (at Ghislaine Maxwell's apartment), in New York, and on Epstein's private island in the US Virgin Islands (in an orgy with numerous other under-aged girls).
"Epstein instructed Jane Doe 3 that she was to give the Prince whatever he demanded and required Jane Doe 3 to report back to him on the details of the sexual abuse.
"Maxwell facilitated Prince Andrew's acts of sexual abuse by acting as a 'madame' for Epstein, thereby assisting in internationally trafficking Jane Doe 3 (and numerous other young girls) for sexual purposes."
The document adds: "Epstein also trafficked Jane Doe #3 for sexual purposes to many other powerful men, including numerous prominent American politicians, powerful business executives, foreign presidents, a well-known Prime Minister, and other world leaders.
"Epstein required Jane Doe 3 to describe the events that she had with these men so that he could potentially blackmail them."
When Epstein was stopped by the FBI, which is understood to have taken a statement from Jane Doe 3, he pleaded guilty to soliciting paid sex with a minor.
According to the court filing, Epstein's lobbying efforts for a favourable plea bargain included "efforts on his behalf by Prince Andrew".
Another close associate of Epstein, who is accused of similar abuse by Jane Doe 3, told the Guardian newspaper that he had never even met the girl in question and that she had made up the claims to "extort money" from Epstein.
Jane Doe 3 told the Guardian in reply: "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."
The Duke's friendship with Epstein, which continued after Epstein had served his prison sentence, caused such controversy that the Duke ultimately stepped down from his role as the UK's trade ambassador.
In 2011 Virginia Roberts, a mother-of-three living in Australia, waived her anonymity to confirm she was one of 40 women abused by Epstein, and claimed at the time that she had also been "sexually exploited by Epstein's adult male peers including royalty".
