Solskjaer, 42, has signed a three-and-a-half year contract with the Tippeligaen club.
He was Molde boss from 2011-2014 before moving to Cardiff City, he failed to keep the Welsh club in the Premier League.
He also had a season as a player with Molde before his 11 year spell with Manchester United in 1996.
Molde have made a good start to their Europa league campaign and are currently sitting two points ahead of the Scottish champions, after a win over Fenerbahce and a draw with Ajax.
In the league are currently seventh in the Norwegian top-flight with 43 points from 27 games.
A statement on the Molde website read: “Molde FK has signed a three-and-a-half-year contract with Ole Gunnar Solskjaer as new manager of the club.
“Ole Gunnar enters with immediate effect and will lead the team against Celtic.”by josh and james