His ex-wife will be known as Sarah Ferguson and no longer Duchess of York, but their daughters will continue to have the title of princess.
He is expected to stay in his Windsor home, Royal Lodge, on which he has his own private lease which runs until 2078.
Prince Andrew has faced intense scrutiny over his links with disgraced financier Epstein, more recently including questions about when he had really cut off contact.
In his BBC Newsnight interview, Prince Andrew had said that he had severed all links with Epstein after they had been photographed together in New York in December 2010.
But emails sent in February 2011 later emerged suggesting that Andrew had privately stayed in touch with Epstein, including swapping a message: "Keep in close touch and we'll play some more soon!"
There had been growing frustration in Buckingham Palace at the scandals that continued to surround the prince.
Next week a posthumous memoir by Virginia Giuffre, who took her own life earlier this year, will be published, which is likely to cast further attention on Prince Andrew's involvement with Ms Giuffre and Epstein.
Ms Giuffre claimed that she was one of many vulnerable girls and young women who had been sexually exploited by Epstein and his circle of wealthy connections.
She alleged that Andrew had sex with her at his friend Ghislaine Maxwell's house in London in 2001, when Ms Giuffre was aged 17.
Her memoir describes two other occasions on which she alleges she had sex with Andrew - in Epstein's townhouse in New York and on Epstein's private island in the US Virgin Islands.
Prince Andrew made a financial payment to Ms Giuffre in an out-of-court settlement in 2022, after she had brought a civil case against him. He denies all the accusations made against him.
Speaking to BBC Newsnight, Sky Roberts, the brother of Ms Giuffre, said Prince Andrew's announcement had brought mixed emotions, but that his late sister "would be very proud" of the development.
"We have shed a lot of happy and sad tears today. I think happy because in a lot of ways this vindicates Virginia," Mr Roberts said.
"All the years of work that she put in is now coming to some sort of justice," he added.
Sky also called on the King to "potentially go ahead and take out the 'Prince' in the Andrew."
Also speaking to BBC Newsnight, Epstein accuser Haley Robson said it was a "bittersweet" moment.
Ms Robson said Ms Giuffre "started this fight and we're going to see it through", adding that "I really wish that she could have been here to see all of what she started".
"Having him [Prince Andrew] step down and be stripped of his titles is something that is long delayed, and it is very appropriate, and kudos to King Charles," Ms Robson said.
Andrew withdrew from Royal Family events last Christmas after an alleged Chinese spy who he was linked to was named in a High Court judgement.
His office said nothing sensitive had ever been discussed with Yang Tengbo, also known as Chris Yang, who became a close advisor on Andrew's business ventures.
In 2022, Andrew was repeatedly named in a court dispute between a Turkish millionaire and her former business adviser.
There were no suggestions of wrongdoing by the prince, but the court case highlighted the prince as receiving payments in a complex set of financial deals. A gift of £750,000 was returned to the Turkish millionaire by the prince.