The fifth-year graduate transfer confirmed his choice Thursday on Instagram. He also was considering Wyoming.
Speight suffered a spinal injury during the fourth game against Purdue last fall and missed the rest of the year. Shortly after the end of the season, he announced that he would not return to Michigan.
In 2016, Speight started 11 games for the Wolverines despite sustaining a shoulder injury. He won the starting job again in 2017 before a broken vertebrae ended the season.
Speight's Michigan career ends with 3,192 yards, 22 touchdowns and 10 interceptions.
UCLA lost a potential transfer earlier this year when K.J. Carta-Samuels changed his mind and opted instead to leave Washington for Colorado State. Speight will be competing with redshirt sophomore Devon Modster and incoming freshman Dorian Thompson-Robinson.
At UCLA, Speight will join former Michigan offensive analyst Jimmie Dougherty, who is now the passing game coordinator and wide receivers coach for the Bruins.
Michigan's quarterback situation also is up for grabs. Ole Miss quarterback transfer Shea Patterson is waiting for an NCAA ruling on whether he would be eligible immediately next season.
Brandon Peters is returning along with Dylan McCaffrey, and Joe Milton will be an incoming freshman.