Lambert, 48, joins the struggling English Premier League side after the sacking of former boss Mark Hughes earlier this month.
Stoke had initially hoped to land Quique Sanchez Flores, the former Watford boss, but were rebuffed over the weekend, before Martin O'Neill also turned down the
chance to take charge in the Potteries.
Lambert, who left Wolves last summer after a short stint at Molineux, is believed to have
greatly impressed club chiefs during his interview at the Bet365 over the weekend, reports the Mirror.
He has signed a two-and-a-half year deal at the club.
Chairman Peter Coates said: “Paul greatly impressed us with his knowledge of our squad and had a clear plan of how he would improve our results.
“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.”
Lambert will officially take charge for the first time on Tuesday, after the Potters' meeting with Manchester United on Monday evening.
Eddie Niedzwiecki will take charge of team affairs, assisted by Kevin Russell and Andy Quy.