Williams, a 6-9 forward, played in 64 games for New York and posted career-best averages of 7.1 points and 3.7 rebounds in 20.7 minutes per game.
"Shawne is a versatile player who will add a needed dimension to our frontcourt," Nets general manager Billy King said. "He can play inside as well as shoot the 3, and is a welcome addition to our roster."
The Nets also waived veteran forward Travis Outlaw and signed veteran free agent forward Ime Udoka.
Outlaw was designated New Jersey's amnesty player in a move that should take roughly $7 million off the Nets' salary cap. The moves came a day after the Nets' bid to acquire Dwight Howard of Orlando fell apart.
Outlaw was signed to a five-year, $35 million contract last summer and appeared in all 82 games, starting 55. He averaged 9.2 points and 4.0 rebounds. The 8-year veteran has played in 482 NBA regular season games with the Trail Blazers, Clippers and Nets, averaging 9.5 points and 3.5 rebounds.
Udoka is a journeyman who has played for five teams in his seven-year NBA career, with his most recent stop being a 20-game stint in San Antonio last season where he had 14 points and 19 rebounds in 130 minutes of action.
New Jersey posted a 24-48 record last season.