A manuscript written by Virginia Giuffre says the Duke had “a really good time” and that she gave him “numerous massages”.
Ms Giuffre – who was 17 or 18 at the time – says she was later rewarded with “close to a thousand dollars” by convicted paedophile Epstein.
The sensational claims emerged in a tranche of newly-released court papers which Epstein’s former lover Ghislaine Maxwell, 58, tried to block.
Describing her time with Andrew at Epstein’s 7,559-acre Zorro ranch near Sante Fe, New Mexico, Ms Giuffre writes: “My job was to entertain him endlessly, whether that meant having to bestow him my body during an erotic massage or simply take him horseback riding.
“The mansion was completely empty save a couple of maids who also cooked our dinners for us, and a couple of bodyguards that we hardly even saw at all.”
She claims Maxwell – a longtime pal of Andrew, 60 – arranged for her to go to the ranch, around 2001, but did not tell her who would be there.
“Ghislaine only told me that I was to meet someone there, not sure of whom that was. It wasn’t my place to ask,” Ms Giuffre claims.
When she arrived at the ranch she saw Andrew who, she says, greeted her with the “same old cheesy
grin” she had seen when they met up previously in London and New York.
Unlike the Prince’s other visits to Epstein’s homes, the American financier and Maxwell were not present.
Ms Giuffre says after two days with Andrew, she was recalled to New York to debrief Epstein and Maxwell.
She says that during a conversation in the tycoon’s kitchen Maxwell asked: “So… how was the ranch with the Prince?”
Ms Giuffre, now 36 and a mum of three living in Australia, says she replied: “I think he had a really good time, he seemed relaxed and when we said goodbye he gave me a kiss.
“I took him horse-riding, just around the property, we went swimming in the pool and, of course, I gave him plenty of massages. He had a massage at least couple of times each day, really seeming to enjoy his time there.”
Describing the reaction of Epstein and his lover Maxwell, she adds: “Like proud parents, they looked me with such content. ‘Good, you did really well,’ Jeffrey complimented me.”
She says Epstein gave her “close to a thousand dollars… more than what I thought anybody at my young age could make for a couple of days work”.
During the Prince’s interview with BBC Newsnight last November, he was at pains to suggest he was never alone while at Epstein’s properties in New York, Florida and the Caribbean.
But he was not asked about visiting Zorro.
A former ranch hand told the Sunday Mirror he saw Andrew a couple of times but never “without Jeffrey or Ghislaine”.
The new claims heap more pressure on the Metropolitan Police to release logs kept by royal protection officers.
One former royal bodyguard, Paul Page, has said notes would include times and dates of trips the Prince took. Logs are kept for 30 years.
Forty-seven court documents were unsealed by New York Judge Loretta Preska on Thursday.
They relate to a 2015 civil case between Maxwell and Ms Giuffre. A second wave of papers could be released – subject to an appeal scheduled for next month.
It was in a 2015 affidavit that Ms Giuffre – referred to as Jane Doe 3 – claimed she had sex with Andrew in Maxwell’s Belgravia home, Epstein’s Manhattan townhouse and at an orgy on the tycoon’s Caribbean island.
Her claims were struck out of the civil case by Florida judge Kenneth Marra.
He ruled: “The factual details regarding with whom and where the Jane Does engaged in sexual activities are immaterial and impertinent to this central claim (i.e., that they were known victims of Mr Epstein and the Government owed them CVRA duties), especially considering that these details involve non-parties who are not related to the respondent Government. These unnecessary details shall be stricken.”
In other claims reported yesterday, Ms Giuffre alleges Andrew had “an orgy with numerous other under-aged girls” at Epstein’s Caribbean home – and Maxwell had “continuous” sex with girls and women.
Maxwell was arrested in New Hampshire last month on suspicion of grooming and abusing three girls under the age of 18.
She is also accused of perjury and faces up to 35 years if convicted.
She has pleaded not guilty and is due for trial in New York in July 2021.
Andrew and his legal team have vehemently denied Ms Giuffre’s claims.
A picture shows them together but the Prince said in his Newsnight appearance that he does not recall ever meeting her.
Epstein, 66, killed himself in jail last year while awaiting sex-trafficking charges.
Andrew was pictured walking with the tycoon in New York’s Central Park in 2010, just months after the paedophile was released from prison for procuring a minor for prostitution.
Andrew told Newsnight: “I went there with the sole purpose of saying to him that because he had been convicted, it was inappropriate for us to be seen together.”
The Duke’s spokesperson declined to comment.
But the new claim was met with shock by his friends.
One said: “The Zorro account is a serious inconsistency. Ms Giuffre’s lawyers have always claimed three alleged encounters. And now this fourth allegation. How so? Why has this never been raised before?
“And where is the evidence? I’d suggest you put these questions to her lawyers, not to the Duke.”