Tony Pulis has offered Woodgate the chance to resurrect his football career but admitted he was taking a chance in signing the injury-plagued defender.
Woodgate has penned a one-year contract after his deal at Spurs expired.
The 31-year-old has undoubted talent but his career has been decimated by injury to the extent that he was able to make just four appearances in the last two seasons at White Hart Lane due to a groin problem.
Pulis conceded his fitness issues are a worry but is confident it is a risk worth taking.
He told the club's official website: "The lad is a top-class player and it's been well documented that he has had his injury troubles, but we believe that if his fitness is managed properly, he will play games for us.
"We know that other Premier League clubs were keen on signing Jonathan, so we are delighted he has chosen us. It's a chance we are taking, but one we were keen to take because of the stature and quality of the player."
He added: "We've effectively taken him on a pay-as-you-play deal which just shows how desperate the lad is to get his career back on the right track."
Stoke chief executive Tony Scholes said: "There is no doubt that we have acquired a player with outstanding quality and experience but he is one who has clearly struggled with injuries over the past two years.
"However, sometimes a fresh start can trigger a change of fortunes. Jonathan has been working very hard throughout the summer on his fitness so he is desperate to make the most of this opportunity and to play in the Premier League."
Woodgate will not fly out to the team's pre-season training camp and will instead meet his new team-mates later this week when they return from Austria.
Woodgate started out at Leeds before spells with Newcastle, Real Madrid and Middlesbrough.
He joined Spurs for £8million in January 2008 but only made 49 league appearances.
He has also won eight England caps, the last of which came in August 2008 against the Czech Republic at Wembley.