The former England U21 goalkeeper was with Preston North End until January this season and arrives at Dens Park having previously played for Everton, Stoke City and Sheffield United.
He joined the League One Lilywhites just before the start of the current campaign, having been released by the Blades last summer.
Simonsen made his first start in the Capital One Cup win over Crystal Palace at Deepdale, going on to make a total of 16 appearances.
His last ended with him suffering a knee injury late on in the Johnstone's Paint Trophy defeat at Coventry City at the beginning of 2013.
The Englishman tweeted: "Now that it's official!! Cannot wait to get going @dundeefconline looking forward to my time up there!!"