Gers boss Walter Smith has been desperate for a goalkeeper during the January transfer window to ensure a back-up to first choice Alan McGregor.
Alexander said: "As soon as I knew Rangers were interested in me, I didn't have to think twice about coming here. It's a fantastic chance to join one of Europe's biggest clubs. I'm grateful to Ipswich for not standing in my way."
Ipswich manager Jim Magilton told the club website: "It's a dream move for him and we couldn't stand in his way. He's back in the Scotland squad and we know he wants to enhance his ambitions.
"It's a great opportunity for him and we wish him well."