The 33-year-old left-back was a free agent following his release by Chelsea this summer and had been linked with various sides, including other Premier League teams as well as overseas outfits.
Cole arrived in Italy on Sunday ahead of his medical before completing the move on Monday afternoon.
The Serie A giants, who finished as runners-up to Juventus last year, confirmed the deal on their official Twitter account.
Tweet: BACKSTAGE: Ashley #Cole's selfie with his new Giallorossi jersey! #HungryForGlory @TheRealAC3
Cole spent eight seasons with Chelsea, during which he won the Champions League, Europa League, Premier League and four FA Cups.
He has 107 England caps and announced his retirement from international football in May when he was left out of his country's World Cup squad.
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