The 21 year old, tipped to have a good future in front of him had refused to sign for newco Rangers, opting to come south of the border to test himself in the English Premier League.
He arrives on a free transfer, the deal is believed to be a four year one once international clearance has been obtained.
Stoke boss Tony Pulis was delighted to get his first summer signing, he told the club’s official website: "Jamie is a lad with enormous potential who is clearly determined to make the most of this fresh start in the game.
"It's important that we introduce some young blood in strengthening the squad and he certainly fits the bill as someone tipped for a bright future."