The 23-year-old enjoyed a successful loan spell with the Championship outfit last season and their bid was accepted a month ago.
Adam had hoped to force his way back into Walter Smith's plans at Ibrox but his five-year spell at the club has now ended.
He becomes the seventh first-team squad member to leave Rangers this summer following the departures of Christian Dailly, Brahim Hemdani, Barry Ferguson, Graeme Smith, Andy Webster and Dean Furman.
A free-kick specialist who scored 18 goals in 86 appearances for Rangers, Adam is expected to go straight into Blackpool's starting line-up for Saturday's Championship opener against QPR.
Coca Cola Ch'ship 09/10 Win Outright: Blackpool 80/1