Law, who played one Test and 54 ODIs for Australia, joined as Bangladesh's head coach in July last year on a two-year contract, replacing compatriot Jamie Siddons. He said that he was stepping down due to family reasons.
Law, who will start in his new assignment after serving his notice period as Bangladesh coach, will assist CA COE head coach Troy Cooley and will be based in his home city, Brisbane.
"The opportunity to return home to be closer with family and work in Brisbane with Australia's established and emerging talent was too good to pass up," Law was quoted as saying in a CA statement.
"I thank the Bangladesh Cricket Board and the Bangladesh players for their support and will be watching their progress with keen interest over the coming months and years," added the 43-year-old former all-rounder.
CA COE manager Belinda Clark welcomed Law's appointment and said his experience of the sub-continent teams and conditions will be of great help for budding Australian cricketers.
"Stuart (Law) is a current international coach with recent demonstrable success and we're fortunate to have secured his services in the Australian cricket landscape," Clark said.
"He has experienced success with two international teams, periods that included an ICC Cricket World Cup final appearance with Sri Lanka in 2011 and Bangladesh's first Asia Cup final last month.