The Lakers formalized the deals Monday.
Ball is expected to make $33.4 million over four years in his contract slot as the second overall pick. The Lakers didn't announce the terms of the deal.
Magic Johnson called Ball "the new face of the Lakers" at his introductory news conference last month. Ball led the nation in assists during his only season at UCLA.
Kuzma is a power forward chosen with the 27th overall selection. The Lakers acquired the pick in a draft-night trade with Brooklyn.
Hart was selected by the Jazz with the 30th overall pick and traded to Los Angeles. He was the Big East's player of the year at Villanova last season.