Last week Scotland international Lee Jones joined the club on loan until the end of the season to provide cover.
Glasgow Warriors head coach Gregor Townsend said: "We're really disappointed for DTH that he's going to be out for the rest of the season."
"He has been an important player for the club over the past few seasons and we wish him well in his recovery."
"We brought in Lee (Jones) last week to provide cover until the end of the season and we were delighted with his display on his debut against Connacht."
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