The Potter has signed Jones in a record-breaking £8million deal from the Black Cats.
Jones has signed a four-year contract at the Britannia Stadium.
“Kenwyne is unplayable on his day, which is why this is a fantastic signing for this football club,” Stoke boss Tony Pulis told the club’s official website.
“We need goals in our team and we also need strong competition among our strikers if we are to continue improving at this level.
“Kenwyne will give us that. It’s no secret that I am one of his biggest admirers, he did a smashing job for us when he was here on loan. He was very young back then, but I believe that he has the best years ahead of him.”
“The fact that I know the manager, some of the players and the place so well has obviously been an influence upon me coming here,” said Jones.
“This is an exciting new chapter in my career and I want to do my best for a club that has clearly had great success since I was last here.
“I am looking forward to the challenge of helping Stoke build upon the progress it has made over the past two years in the Premier League.”
