Lampard, taking on his first managerial role, was one of 20 applicants for the job and becomes Derby's seventh appointment in just over three years.
He replaces Gary Rowett after he left to join fellow Championship side Stoke.
"I've always wanted to manage a club with a big tradition and history like Derby County, so this is a huge opportunity," said the 39-year-old.
"I have spent considerable time discussing the role and the club's objectives with the chairman and board members.
"This is my first job as a manager, but I've worked closely with some of the best coaches in the game and I'm confident in my own abilities and those of the team around me, including the board.
"I know it won't be easy - managing a football team never is - but I'm really looking forward to the challenge ahead."
Lampard, capped 106 times by England, made 649 appearances for Chelsea and is the club's record goalscorer.
Derby finished sixth in the second tier last season but lost in the play-off semi-finals to Fulham and have been without a manager since Rowett's departure on 22 May.
Chelsea youth team coach Jody Morris, a former Blues team-mate of Lampard's, is expected to join the Rams' coaching staff.
Lampard left Chelsea in 2014 having won 11 major trophies. He has three Premier League titles, four FA Cups, two League Cups, the Europa League and the Champions League from his time in west London.
He started his career with West Ham, moving to Chelsea for £11m in 2001 before later going on to play for Manchester City and New York City prior to his retirement in February 2017.
Following his international debut in 1999, Lampard scored 29 goals for England and appeared in three World Cups and one European Championships.
"Few players have achieved what Frank has in his career to date," said Derby chairman Mel Morris. "He's a winner, a leader who knows what it takes to succeed and who has the character and charisma to be a fantastic manager for us.
"The board were unanimous that he was the stand-out applicant with many highly desirable and unique capabilities."