The Scotland Under-21 midfielder is a free agent after exercising his right not to have his Rangers contract transferred to the 'newco' formed in the wake of the Glasgow giant's financial problems.
"Jamie is a lad with enormous potential who is clearly determined to make the most of this fresh start in the game," manager Tony Pulis told stokecityfc.com.
"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."
Ness made just 18 appearances for Rangers but missed most of last season with a groin injury.
Rangers has threatened legal action against players who refuse to transfer their contracts to a newly-formed Ibrox club and has also suggested it will pursue clubs for compensation for players such as Ness.
PFA Scotland believes all players who have terminated their contracts are legitimate free agents and have asked the football authorities to release their registration from Rangers, but newco chief executive Charles Green has vowed to seek recompense.
Scottish Premier League
Sheffield United has made an approach to Motherwell for permission to hold talks with Stuart McCall over its managerial vacancy.