Having been released by Chelsea, the former England international had been linked with a host of clubs around the world.
However, earlier this month it was confirmed that the 33-year-old had agreed a two-year contract at the Stadio Olimpico.
Speaking for the first time about his decision, Cole told reporters: "I decided to come here in order to continue to play in the Champions League at a great and ambitious club. This is a great challenge and I am ready, I am ready to fight for a place in the team and to try to win."
Cole has been handed the number three shirt by La Maggica.