The 29-year-old was stretchered off the Murrayfield pitch with an oxygen mask in the 30th minute of his country's 23-17 loss to France and the initial estimates of his recovery time have been placed at threeto four months.
Lamont has only made one RaboDirect PRO12 appearance for his club since signing for Glasgow in December from French club Toulon.
But his absence will still be a blow to the Warriors as they seek to fend off the challenges of Ulster, Cardiff Blues and the Scarlets to claim fourth and a place in the play-offs in May.
"He (Lamont) underwent surgery on Monday lunchtime to stabilise a fracture to his fibula," said Scotland team doctor James Robson.
"His convalescence will get underway at once. Injuries of this nature require typically three to four months of rehabilitation."
Meanwhile, Warriors coach Sean Lineen was lamenting the injury, but has backed Lamont to make a full recovery.
"On behalf of everyone at Glasgow Warriors, I'd like to wish Rory all the best with his recovery from this injury," Lineen told glasgowwarriors.org.
"He's had bad luck on the injury front on a number of occasions in the past, and he's always shown the strength of character to come back strongly.
"He's in great hands with both the Warriors and Scotland medical teams."
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