She alleges she was forced by financier Jeffrey Epstein to have sex with the prince when she was a minor in the US.
In the latest papers filed in Florida, the US woman, known legally as Jane Doe #3, said she knew the prince was a member of the British royal family but that she just called him ?Andy?.
Buckingham Palace has already strongly denied the allegations.
They said on Wednesday they had nothing to add following the filling of the new court papers.
This is the first time the woman, named elsewhere as Virginia Roberts, has given a statement in her own words to the courts.
Her lawyers said they had sent Prince Andrew a letter asking him to respond under oath to the allegations.
They say the letter was sent back to them as having been refused by the recipient.
?Hurt by denials?
In the letter they wanted to discuss the circumstances around a photograph of Jane Doe #3 and Prince Andrew, as well as the other times she says they met.
The woman says in the new court papers that Epstein had taken the photograph ? which appeared in the latest court documents.
BBC Washington correspondent Rajini Vaidyanathan said of the fresh documents: ?Over several pages, and crucially, in her own words, Jane Doe #3 reasserts her claims that she was forced to have sexual relations with Prince Andrew on three separate occasions, including once at an orgy.?
In the documents, the woman claims that Epstein had introduced the pair, telling her she was to meet a ?major prince?.
She has also alleged that Epstein told her that whatever Prince Andrew wanted, she was to make sure she got.
Our correspondent added: ?Her lawyers say she has taken this step because she stands by her story, and has been hurt by denials from Buckingham Palace.
?Prince Andrew has yet to comment on this latest development, but has in the past has strongly rejected the allegations, describing them as false and without foundation.?
The allegations against Prince Andrew emerged in evidence submitted to a US court in a case involving US financier Epstein, who spent time in jail in 2008-9 for a sex offence with a minor.
Prince Andrew, 54 and fifth in line to the throne, stepped down as UK trade envoy in July 2011 following controversy over his friendship with Epstein.
The prince is due to make a public appearance at the World Economic Forum at Davos on Thursday but Jane Doe #3?s lawyers said the filing of her statement was not timed to coincide with that.
Prince Andrew is not party to any legal proceedings.