Campbell suffered the injuries when Washington Capitals superstar Alex Ovechkin sent him into the end boards on Sunday.
"He will likely avoid surgery and we anticipate a full recovery," team physician Dr. Michael Terry said in a statement. "He should be ready to play in approximately seven to eight weeks."
Ovechkin was suspended two games by the NHL, which called his hit "reckless." He received a five-minute boarding penalty and a game misconduct for the hit.
Ovechkin issued this statement through the Capitals on Tuesday:
"I am very sorry that Brian was injured and I hope he is able to return to his team soon. NHL hockey is a physical game. We all play hard every time we are on the ice and have battles each shift in every game we play so we can do our jobs and win. As players we must accept responsibility for our actions and I am no different, but I did not intend to injure Brian and that is why I was disappointed with the NHL's decision [on Monday].
"Every time I have the honor to play for my team, I will continue to do what I have done since I was taught to play. I will play hard, play with passion and play with respect for my teammates, opponents and fans. I look forward to returning to my team and doing everything I can to be the best player I can be."
The Hawk recalled Nick Boynton, acquired from Anaheim before the deadline for future considerations, to take Campbell's place in the lineup.