Britton, a second-round draft pick in 2009 who has started 22 games, tore his right labrum Sunday at Kansas City.
Britton dislocated his shoulder in the second quarter, popped it back in place himself and returned to the game. He dislocated it again on the next series. This time, he couldn't get it back in place without help from team doctors.
He believes "the damage was done at that point."
Tests Monday revealed the extent of the injury. He plans to have surgery soon.
Jordan Black, a seventh-year veteran, will replace Britton in the lineup beginning Sunday at Dallas.