The 48-year-old succeeds Mark Hughes, signing a two-and-a-half-year deal at the 'Potters'.
Stoke currently sit 18th in the Premier League and are out of the FA Cup after being humbled by League Two side Coventry City in the Third Round, the result which ultimately led to Hughes' departure.
Lambert will watch Monday night's game against Manchester United from the stands at Old Trafford before taking charge on Tuesday.
The Scot's first match in charge will be the home clash with fellow strugglers Huddersfield Town on Saturday.
Lambert has been out of work since leaving Wolves at the end of last season, but has been handed a shock return to Premier League management after Stoke missed out on their top three targets.
Espanyol boss Quique Sanchez Flores opted to stay at the Spanish club, Republic of Ireland manager Martin O'Neill snubbed interest from the club and Derby manager Gary Rowett signed a new deal to commit his future to the Pride Park club.
Stoke chairman Peter Coates was nevertheless delighted with the appointment, saying: "Paul greatly impressed us with his knowledge of our squad and had a clear plan of how he would improve our results," he said.
"He's a man who backed himself as a player, none more so than when he turned down contract offers in Scotland to go on trial in the German Bundesliga with Borussia Dortmund, and it's obvious he adopts the same approach as a manager."
15 January 2018