The Welsh flanker's injury occurred after his strong tackle on Scottish centre Max Evans in their thumping 51-3 over the 14 man Scotland side.
Warburton is said to be out for around several months as the 25-year-old will have to undergo surgery in the coming weeks.
The Cardiff born flanker also features for Cardiff Blues in the RaboDirect Pro12, who stated on their club website that Warburton will be having surgery.
A statement confirmed: "Following MRI scans and consultations with specialists, Warburton will have surgery and is expected to miss several months of action."
The regular for the Cariff Blues has only played six games this season; having played four games in the league and twice in the Heineken Cup.
Wales were hoping to retain the Six Nations title this year but after two defeats to both England and Ireland, their dream was looking bleak.But with solid performances against France, Italy and Scotland; Wales finished third above France on points difference.
But joining Warburton on the injury list for Cardiff Blues is full-back Leigh Halfpenny, who missed the Scotland encounter due to surgery.
Cardiff Blues responded by wishing both players a speedy recovering: "Everybody at Cardiff blues wishes Sam and Leigh all the best during their rehabilitation."
The Blues failed to make the knock-out stages of both the Heineken Cup and LV Cup, as Dale McIntosh and Paul John's side sit tenth in the Pro 12 with six games remaining; including encounters against Welsh rivals Ospreys and Scarlets.
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