The 32-year-old, who won his one and only international cap against England in 2008, will officially join the Hoops when the transfer window opens on January 1.
Compper flew into Scotland on Tuesday and terms have now been finalised, making the former Fiorentina and Hoffenheim defender the third German to link up with Celtic after Andy Thom and Andreas Hinkel.
Speaking to his new side's official website, Compper said: "When I first heard of Celtic's interest I was immediately very keen and eager to come. They are a club who play for titles and that is something which has been lacking throughout my career.
"I was playing in good teams but never won a title. This is something I'm really looking forward to, having the possibility to play in big games for big wins.
"Celtic is a club that is always going for the big wins. I've played in important games in Italy and Germany. I was in a cup final in Italy and we lost, unfortunately, but I hope I can add silverware to my career here."
Compper will be in attendance at Celtic Park on Wednesday evening for the Bhoys' meeting with Partick Thistle.