"It’s fantastic news that we have been able to push through the deal," said Potters boss Tony Pulis. "Brek is a very, very exciting player, and one who we firmly believe could go on to become an outstanding player for this football club.
"He has terrific pace, a very good left foot and is capable of scoring goals. He has all the tools needed to be a success here."
Shea has played for Dallas since 2008 and is recovering from surgery in November on his right foot. He has yet to score after 15 appearances for the U.S. national team.
Shea had already passed a medical with Stoke but the deal was held up as he does qualify for a work permit under current regulations.
However, that was largely due to Shea undergoing an operation last August to remove a bone in his right foot, forcing him to miss several internationals, and Stoke successfully argued mitigating circumstances.
"It’s been up and down, trying to figure out what’s going on, but I’m here now and ready to go," said Shea. "It’s one of the best Leagues in the world, and I’m excited to be here and hopefully I can contribute."