By Andrew Tuft
Adam has signed a four-year contract at the Britannia Stadium after spending only a year at Anfield following a move from Blackpool a year ago.
The fee for the Scotland international is officially undisclosed.
The 26-year-old completed a medical with the Potters on Friday and agreed personal terms with the club in time to be registered as a Stoke player ahead of the weekend’s Premier League fixtures.
Stoke play Wigan Athletic at the DW Stadium on Saturday.
Adam began his career in Scotland with Rangers, making 87 appearances for the Ibrox side and spending loan spells with Ross County and St Mirren.
Adam moved to Blackpool in 2008-09, initially on loan, before making the move permanent and helping the Tangerines to the Premier League in 2010-11. In total, Adam made 95 appearances for Blackpool before moving to Liverpool.