Since fermented beer contains alcohol, it's freezing point will be lower than water. How much lower depends on how much alcohol is in the beer. For most mid-range beers, you can assume they will freeze in the 29ºF range to be safe.
Keeping your beer in a keg at 32ºF is probably fine for any beer, and necessary for many... The process of lagering is exactly this (keeping it at 32-34ºF for a several weeks).
If you do not have an external temperature control, a freezer will freeze your beer. If you used a refrigerator, you should be able to set it to 36-38F without an extra control.