Kenneth, who had missed the opening game of the League Two campaign against Oxford after suffering a hamstring problem during a pre-season friendly, was set to make his debut for the Pirates at Barnet on Tuesday night.
However, the 25-year-old suffered a fresh injury problem during the warm-up.
"I'm gutted for him," Rovers manager Mark McGhee told BBC Radio Bristol. "We know Garry's general condition hasn't been great as he hadn't been training all summer. But you could tell how upset and disappointed he was.
"During that six weeks we're going to work as hard as we can to get him as fit as we can, so that he stops having these unnecessary injuries."
Kenneth had spent his entire career with Dundee before moving to the Memorial Stadium on a three-year deal this summer.