Bauer agreed terms and had a medical last week but has now completed the move to Stoke, who sacked Hughes on Saturday after a shock FA Cup third round defeat by fourth-tier Coventry City.
“Moritz is a player we have been trailing for some time and was someone Mark Hughes wanted to bring into the club," Chief Executive Tony Scholes said on the club's website.
“Moritz came over to England last week to undergo his medical and discuss terms and we have now been able to conclude his transfer."
Stoke are in the relegation zone with 20 points from 22 matches and visit Manchester United on Monday. They will hope Bauer can help tighten up a defence that has by far the worst record in the Premier League this term with 47 goals conceded.
Although Bauer represents Austria, he was born in Switzerland and started his career with Grasshoppers Zurich, making more than 90 league appearances.
He joined Rubin Kazan in 2016 and played 37 league games for the Russian Premier League side.
(Writing by Ken Ferris; Editing by Nick Mulvenney)
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