Scotland wing Lamont was injured during the first half of Scotland's 23-17 defeat by France in Sunday's Six Nations international at Murrayfield when, after jumping to contest a re-start with French No 8 Louis Picamoles, he landed awkwardly.
After receiving treatment for several minutes, he was stretchered off.
Glasgow's Lamont has since undergone surgery and Scotland team doctor James Robson told www.scottishrugby.org on Tuesday: "Rory is comfortable. His convalescence will get underway at once.
"Injuries of this nature require typically three to four months of rehabilitation. We all wish Rory a speedy recovery."
Lamont was replaced by Nick de Luca at Murrayfield, where Scotland, despite a much improved display, suffered their third straight defeat in as many Six Nations matches this season and fifth loss in a row in total.
Scotland conclude the Six Nations with potentially tricky away fixtures against Ireland and Italy next month.