The 39-year-old joins on a two-year deal after he was released from the final year of his contract at their Midlands rivals.
Given has won 128 international caps and has turned out for clubs including Celtic, Manchester City and Newcastle during a 21-year professional career so far.
The move to Stoke will see Given once again link up with Potters boss Mark Hughes, who signed the Irishman when he was in charge of City in 2009.
Hughes said: “I know Shay particularly well from our time together in Manchester and I’m really looking forward to working with him again.
“He will not only be a great asset on the pitch as an outstanding goalkeeper, but he’s also a fantastic character to have in the dressing room and around the club – a true professional.”
Given said he was also keen to link up with Hughes again and join a growing Irish contingent at the Britannia.
He said: “Hughes was a big pull for me to join. He’s done a magnificent job and I have a lot of respect for him. He’s a man who instantly gets the respect form his players.
“I know the Irish lads really well obviously, so that will be a big help for me to settle in the dressing room.”
Given’s move to Stoke should pave the way for current goalkeeper Asmir Begovic to replace Petr Cech and join Chelsea in an £8m deal.
Stoke have already signed Chelsea midfielder Marco van Ginkel on a season-long loan, while England full-back Glen Johnson is understood to be undergoing a medical at the Britannia Stadium ahead of joining the club on a free transfer from Liverpool.