The Munster number 10 did the damage during a 38-17 Pro14 defeat to Ulster last week and will be out for up to four months.
Carbery was making his first start of the season after recovering from an ankle injury, but now faces a lengthy absence.
Munster head coach Johann van Graan said: "That's rugby, I'm gutted for Joey as an individual.
"He worked hard to get back into the position, he played the full 80 for the first time for quite a while and now he is out for a considerable amount of time.
"That's rugby, that's life. He'll be back, he's a class man and a brilliant rugby player.
"He'll come back stronger, take his time and I can't wait to have him back in the future."
Carbery posted on Instagram: "Devastation doesn't even describe how I'm feeling. Thanks for all the well wishes. Been a tough couple of months physically and mentally, and thought I was in the clear. But will be back soon, better than ever."
Ireland's first-choice five-eighth Johnny Sexton is also out of action with a knee problem and is reportedly unlikely to feature before Ireland's Six Nations opener against Scotland at Aviva Stadium on 2 February (AEDT).