Gudjohnsen, capped 84 times, made three appearances in qualifying and scored one goal as his country reached the finals of a major tournament for the first time.
"Obviously I want to play for the national team, these are exciting times for Iceland," the 37-year-old told the Verdens Gang newspaper on Friday.
"But I didn't sign only for that. I have had many good references about Molde and I still love playing football."
Gudjohnsen, who spent last season at Chinese side Shijiazhuang Ever Bright, will boost Molde's resources after their disappointing sixth-place finish in the top flight in 2015.
"It's fantastic for us as a club to have signed such an experienced player and such a good player as Eidur," said coach Ole Gunnar Solskjaer in a statement.
Former Manchester United striker Solskjaer is in his second spell in charge of Molde after winning back-to-back titles with them in 2011 and 2012.
Gudjohnsen won two Premier League titles with Chelsea and the Champions League and La Liga with Barcelona. He also landed the Dutch top-flight crown with PSV Eindhoven.
Euro 2016 in France starts in June.
(Reporting by Philip O'Connor; Editing by Tony Jimenez)
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