The Iceland international chose the Potters ahead of several rival English offers.
Gudjohnsen commented: "I am delighted to be here, it is a new challenge, indeed a big challenge, but one I am really looking forward too.
"After speaking to the gaffer and hearing his and the club's ambitions, he really swayed my mind, and I am delighted the deal is done.
"I can't wait to step on to the pitch, the fans are fantastic here and I always remember the atmosphere from when I have played here before.
"I scored here not so long ago for Tottenham, so maybe I owe the supporters something. Hopefully I can reward them with some goals."
He added: "I am an ambitious player, but this is an ambitious football club. I want to help take the club on to the next level. That is my main objective."