1993 Buick Century rear shock mount

Depends on what's broken and why. If it's due to metal fatigue and something cracked, it can usually be welded. If it's due to rust, the part needs to be replaced. Very often on GM cars that means replacing a rear lower control arm. The car will need to be aligned after that. It's very important to not tighten the pivot bolts until the car is sitting on the tires and is at normal ride height. If you tighten the bolts with the suspension hanging down, when you lower the car off the jack stands you'll put the rubber bushings in a permanent twist which will greatly shorten their life.

If you have a solid beam axle, there's a shock absorber mounting bracket on it. Those can be ground off and replaced as long as the rest of the axle is solid and not rusty.