The 25-year old has put pen to paper on a three-and-a-half year contract and the eventual fee could increase by £250,000 if Palace are promoted.
Danns joined Birmingham from Colchester in the summer of 2006, but never established himself as a regular first-team player at St Andrews.
However, he was a star performer during his time at Colchester - scoring an excellent 28 goals in 87 appearances from midfield.