Many black bears have been killed with .30-30s around here. I even know of a couple that were killed with a .357 magnum handgun. I would bet that almost any .308 round will be just fine.
.308 is a great round for black bear. What weight bullet you need would depend on the size of the bear you end up shooting. A 180 grain Nosler Partition would work on even the largest black bear you would encounter.
I've had my best luck shooting Remington 180 grain core-lokt's in my .308 carbine. The roundnose style opens up fast like a speedier 150 grain but has the weight and momentum of the 180 grain.