The duchess said in a statement that she had financial problems, but "that is no excuse for a serious lapse in judgment and I am very sorry that this has happened."
"I very deeply regret the situation and the embarrassment caused," she said.
The tabloid News of the World posted video on its website that appears to show Ferguson discussing payment terms. She is heard to say "500,000 pounds when you can, to me, open doors."
Asked if she was referring to the prince, she said: "Yeah."
The newspaper said Ferguson, 50, spoke to an undercover reporter posing as a businessman.
Andrew is an international British trade ambassador. Both the newspaper and the duchess said he had no knowledge of the meeting.
Buckingham Palace also said the prince had been unaware of the meeting. It said he carried out his trade role with "complete and absolute propriety and integrity."
The tabloid report is embarrassing for Ferguson, but there is no suggestion she did anything illegal.
Ferguson married Andrew, who is fourth in line to the throne, in 1986. They had two children, princesses Beatrice and Eugenie, before divorcing in 1996.
Since then Ferguson has written children's books, made television documentaries and acted as a spokeswoman for Weight Watchers.
She also has struggled financially and has complained that her divorce settlement was stingy.
Recently the company set up to manage her U.S. career in publishing, public speaking and media work, Hartmoor LLC, collapsed with debts of around $1 million.
Ferguson is scheduled to receive an award for her charity work in Los Angeles on Sunday.
The sting was the latest scoop for News of the World undercover reporter Mazher Mahmood, nicknamed the "Fake Sheik" after his signature ruse of pretending to be a rich Gulf businessman to dupe celebrities, politicians and suspected criminals.
He is a controversial figure who has targeted members of the royal family before. One of Mahmood's most famous scoops was a 2001 sting involving Prince Edward's wife. Posing as an aide to a Saudi Arabian prince interested in hiring her public relations company, Mahmood charmed Sophie, the Countess of Wessex, into making indiscreet comments about the British government. She also was caught on tape describing then-Prime Minister Tony Blair's wife, Cherie, as "horrid, horrid, horrid."
