The 32-year-old Denmark international has been on trial with the club recently and was available on a free transfer after being released by Villa at the end of last season.
He becomes Tony Pulis' third recruit ahead of their maiden Premier League campaign, following Dave Kitson and Seyi Olofinjana.
Pulis hailed the move as "superb business". He said: "Thomas is a fantastic acquisition for this football club.
"He has played almost 400 games in the Premier League and been number one for Denmark for a very long time.
"The fact that we have got him on a free transfer too is superb business for us, because I believe he is a very, very good goalkeeper."
Potters chief executive Tony Scholes said: "The club are delighted to have attracted a player of the quality and calibre of Thomas.
"He becomes our third major signing of the summer following on from Dave Kitson and Seyi Olofinjana.
"Thomas is still the Danish number one, and is a player who possesses bags of top-flight experience."