Williams, taken second overall by the T-Wolves in the 2011 NBA draft, has averaged 10.1 points and 4.9 rebounds during his two-plus seasons with the club.
The natural power forward was switched to small forward but has seen only limited playing time for Minnesota this season.
In coming to the Timberwolves, the 6-foot-8 Mbah a Moute joins former UCLA teammate Kevin Love.
He has averaged 6.8 points and 5.3 rebounds in 344 career games with the Milwaukee Bucks and Kings.
"We are excited to acquire a solid veteran player in Luc Mbah a Moute," Timberwolves President of Basketball Operations Flip Saunders said. "Luc is known as one of the premier defensive players in the league with an ability to guard multiple positions."
The trade came as the T-Wolves have lost five of seven games following a 4-2 start.
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