Welsh is expected to be unavailable for selection for a minimum of six months.
Head coach Gregor Townsend told www.glasgowwarriors.org: "It's clearly disappointing for Jon and the club as a whole that he will miss a substantial amount of rugby in the first part of the season.
"He will receive excellent care and attention from our medical team as he recovers from the injury, and we look forward to welcoming him back to the fray in due course.
"On behalf of everyone at the club, I'd like to wish Jon all the best with his rehabilitation. He's a positive character, which will stand him in good stead for the challenges ahead."
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