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Former Highlanders coach Brown will take charge of the struggling Super Rugby outfit after Joseph announced he would focus on his role as coach of the Japan national team heading into next year's World Cup.
Brown, who also succeeded Joseph as coach of the Highlanders in 2017, is the fourth former All Black to head the Sunwolves in four seasons. Mark Hammett and Filo Tiatia previously held the role.
Brown took charge of the Sunwolves in the latter part of this year when a back ailment forced Joseph into a period of recovery.
It didn't help the team, who picked up the wooden spoon for the second time.
Another Kiwi, Scott Hansen, will be a Sunwolves assistant next season and will take charge when Brown is periodically required in his other role, as an assistant to Joseph with Japan.
Sunwolves chief executive Yuji Watase expected the changes to be seamless.
"Tony is a top-class attack coach and both the Sunwolves and the Japan national team continue to benefit from his coaching," he said.
"Jamie and Tony have worked closely for a number of years and we are confident there will be continuity in approach and vision."