Cook, 54, has signed a contract until the summer of 2023 and inherits a team that is eighth in the table, two points off the play-off places.
He has been out of work since leaving Wigan in July, when the Latics almost avoided relegation from the Championship despite a 12-point deduction for going into administration.
Cook's appointment comes with US investors reportedly close to buying Ipswich from long-time owner Marcus Evans, with Lambert openly stating his departure was a result of the impending change in ownership.
"First I want to wish Paul and (former assistant) Stuart Taylor all the best for the future. I know them both and it’s a hard business, football," said Cook.
"I’m proud to be in this position and I’ll be doing everything possible to repay the faith Marcus has shown in me. Winning promotion this season is the immediate target. It will be a big challenge but it’s a challenge that everyone at the club has a part to play in.”
During highly successful spells with Wigan, Portsmouth and Chesterfield, Cook has won both the Sky Bet League One and League Two titles.
Ipswich owner Marcus Evans told the club’s official website: "I’m delighted to announce that Paul Cook is our new manager.
"He has a proven track record in getting clubs promoted and importantly, not only knows the challenges League One presents but also has experience of working in the Championship and getting back to that level remains our primary focus."
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