Virginia Giuffre says she was brought to London for sex with Prince Andrew in March 2001, when she was 17 years old.
Prince Andrew says of the claim he had sex with her he can "absolutely and categorically" say "it never "happened". He says he has no recollection of ever meeting her.
Ms Giuffre's first interview for British television has been given to the BBC as part of a special hour-long Panorama.
It includes an account of how she was trafficked to London by the sex offender and financier Jeffrey Epstein.
Her claims have put the sex life, judgement and honesty of a senior member of the Royal Family under intense scrutiny.
Buckingham Palace says the Duke of York "emphatically denies having any form of sexual contact or relationship with Virginia Giuffre and that any claim to the contrary is false and without foundation."