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The 48-year-old joins the team after spending the past five seasons as an assistant at the Toronto Raptors.
"Adrian is a widely-respected coach and former player, who brings great leadership and experience to our team," said Bucks general manager Jon Horst.
"His championship-level coaching pedigree, character, basketball acumen and ability to connect with and develop players make him the ideal choice to lead our team. He has earned this opportunity."
Griffin's appointment follows Milwaukee's stunning first-round exit from the Eastern Conference playoffs which led to the dismissal of Budenholzer on May 4.
"The Bucks are a championship organization and I'm thrilled to work with an established roster of high character and talented players," Griffin said in a statement.
"I'm excited to be back in Milwaukee and I can't wait to get started."
Griffin began his coaching career in the NBA at the Bucks in 2008, where he worked as an assistant to then head coach Scott Skiles.
The Bucks finished the 2022-2023 regular season with the league's best record of 58-24, but were eliminated in five games by the eighth-seeded Miami Heat in the first round of the playoffs.
Budenholzer, a two-time Coach of the Year who guided the Bucks to their first NBA title in 50 years in 2021, was subsequently dismissed.