Breeze, who has played 46 times for Blues and scored two tries, will have surgical repair this week before commencing rehabilitation as he looks for a playing return at the start of next season.
Director of Rugby Phil Davies said: "You have to feel for Marc because he has played really well this season and has been an important part of our squad. It's a huge blow for him and us as it takes us down to two fit hookers.
"We will now make sure he gets the very best treatment and rehab to make a full and successful recovery and return to playing as soon as he is fit."
Davies also revealed contrasting news in the rest of his squad with Owen Williams likely to miss the Glasgow game but Wales ace Cory Allen among those back training.
He added: "Owen has got a little bit of a tweak with his hamstring and because we have got a Saturday and Thursday turnaround that is going to be tough to get him in.
"We are fortunate we have Gavin Evans, Dafydd Hewitt, Richard Smith and Isaia Tuifua so are not too badly off in the centre position.
"We have got to make sure that Owen recovers properly. He has been out for too long for us to push, but we are hoping he could be involved next Thursday.
"Rory Watts-Jones is back in training and Cory Allen and Josh Navidi are both back non-contract training, so not yet available for selection. Slowly we are getting a few back but also losing a few. We will manage as best we can and take it a game at a time."