Cort has been on loan at the Britannia Stadium since November, making 15 appearances and scoring four goals to help the club into the play-offs.
He had been named as the Eagles' player of the year last season but has now ended his 18-month association with the club by signing a three-and-a-half year deal with Stoke.
If promotion is achieved, then Cort's fee will rise by a further £200,000 due to a clause added by Palace.
"This club record signing shows the club's determination to build a team strong enough to challenge for promotion," Scholes told Stoke's official website.
"He's a vastly experienced defender who's done extremely well for us while he's been on loan.
Stoke boss Tony Pulis also expressed his satisfaction at the signing of Cort saying: "I'm delighted we've signed Leon permanently.
"Since he's come in he's been a smashing lad on and off the pitch who's a real danger at set-pieces."