The move comes after Laws signed former Manchester City goalkeeper Nicky Weaver earlier on Wednesday.
Cort was Stoke's record signing at £1.2million when he arrived at the Britannia Stadium from Crystal Palace two years ago.
He played a key part in the club's promotion to the Premier League in the 2007-08 season.
However, he has found his first team opportunities limited since then due to the form of Ryan Shawcross and Abdoulaye Faye, not to mention summer signing Robert Huth.
Stoke manager Tony Pulis said: "Leon was an influential figure in the club's promotion triumph and a dependable performer whenever called upon in our first season in the Premier League.
"We would like to thank him for his contribution to that success and wish him all the best in the future with his new club."
With his side in a scrap to avoid a quick return to the Championship, Laws was determined to strengthen his squad.
After losing his first three games in charge since replacing Owen Coyle, now at Bolton, he said: "The one thing the players need is support.
"We haven't got the biggest squad in the world and when you lose good players it affects your team."
Laws expects to call on defender Andre Bikey for the home match against Chelsea on Saturday.
It is a boost for Laws as influential skipper Graham Alexander (calf) and Chris McCann (knee) were both injured in the first 15 minutes against Bolton.
So the return of the versatile Bikey, who has been on African Nations Cup duty for Cameroon, could not be more timely.
Laws said: "We are due to assess the injuries to Graham and Chris. But the good thing is we have got Andre back from international duty, so that is a plus.
"I spoke to him on the phone, so we know he will be here for the weekend. That is great news because we need all the quality we can at this time."
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