The club revealed the 21-year-old underwent successful surgery on a fracture to his right fibula over the weekend and has now been discharged from hospital.
Head coach Gregor Townsend said: "We're all gutted for Duncan. He's had a really good season and he'll be disappointed to miss the final few games. He's now on the road to recovery and we wish him all the best for his rehabilitation."
Weir added: "I'd like to thank everyone for their messages of support over the weekend. My phone has not stopped and it's really helped me over the last couple of days."
- Sports & Recreation