Andrew has faced weeks of scrutiny over his friendship with Epstein, who committed suicide in a Manhattan jail cell earlier this month while being held on sex-trafficking charges.
Andrew, who is the second son of Queen Elizabeth, issued a lengthy statement saying he wanted to "clarify the facts" around his relationship with Epstein.
"At no stage during the limited time I spent with him did I see, witness or suspect any behavior of the sort that subsequently led to his arrest and conviction," Andrew said.
British media including the Daily Mail have published a picture which they said showed Andrew waving goodbye to a woman from inside a Manhattan mansion owned by Epstein.
The Mail said the picture had been taken in 2010 - two years after Epstein pleaded guilty to a Florida state felony prostitution charge and registered as a sex offender.
U.S. court papers have previously shown that Epstein had socialized with Andrew and other high-profile figures including U.S. President Donald Trump and former president Bill Clinton.
Andrew, 59, said it was a "mistake and error" to see Epstein in 2010 after he pleaded guilty to paying a teenage girl for sex.
He said that he first met Epstein in 1999, saw him once or twice a year and stayed in a number of his properties.
"His suicide has left many unanswered questions and I acknowledge and sympathize with everyone who has been affected and wants some form of closure," said Andrew, whose title is the Duke of York.
"What I thought I knew of him was evidently not the real person, given what we now know."
His statement on Saturday was issued just days after Buckingham Palace issued a statement in which Andrew rejected any suggestion that he participated in the alleged sex crimes that Epstein was accused of.
Epstein first came under investigation in 2005 after police in Palm Beach, Florida, received reports he had sexually abused underage girls in his mansion there.
By 2007, Epstein was facing a potential federal indictment for sexually abusing dozens of girls between 1999 and 2007. Epstein struck a deal, however, to plead guilty in 2008 to the Florida state felony prostitution charge, and register as a sex offender.Epstein, 66, was arrested on July 6 and pleaded not guilty to federal charges of sex trafficking involving dozens of girls as young as 14.