Priestland, 25, was carried off the pitch during Scarlets's 22-16 Heineken Cup defeat against the Exeter Chiefs on Saturday, and had surgery on Tuesday.
He is expected to be out of action for at least six months.
"It's a big blow for Rhys and for our region to lose a player of his quality and influence in our team," Scarlets head coach Simon Easterby told the club's website.
"He was coming through well in the last couple of games of the autumn, put in a strong performance against Australia and was making a real impact in our game against Exeter - so it's very difficult for him to take; but he'll get plenty of support from here throughout the injury period."
Wales begin their Six Nations title defence against Ireland at the Millennium Stadium on February 2 as they look to end a run of seven consecutive Test defeats.
The Lions play their first match against the Barbarians in Hong Kong on June 1, before heading to Australia for nine matches, including three Tests.