The 31-year-old Brazilian, who scored the extra-time winner to knock the Hoops out of the Champions League at the last 16 stage in 2007, can't wait to get back to his San Siro stomping ground.
Even though he cost £56million in 2009, Kaka found game time at a premium while Jose Mourinho was manager of Real Madrid.
And the arrival on Monday of Gareth Bale for a world-record fee helped convince him he was not going to be a first pick even with the Special One having moved to Chelsea.
Kaka wants to make up for lost time and explained as he arrived in Milan: "Four years have passed - and now I'm back.
"It's been two weeks that I thought, if I could come back, a return to Milan would be ideal for me.
"In the last few days, I've been dreaming of hearing the fans sing my name in the San Siro again."
The 2007 Ballon d'Or winner is understood to have made the move on a free transfer, and a statement by the club read: "Adriano Galliani sealed the deal to bring Kaka back to Milan between 2.30am and 3am this morning.
"The player has been given a contract ending on June 30, 2015."
Celtic open their group stage campaign against Kaka's Milan at the San Siro on September 18, and the Serie A club travel to Parkhead on November 26.