In Alaska it is illegal to shoot a cub or a sow with cubs. The definition here of a black bear cub is a bear in its 1st year of life. That means if it was born in the spring, it would not be legal until the next spring season. It is pretty obvious to tell the difference between a first year cub, and a 2 year old bear.

I'm almost certain that any hunter would tell you not to shoot a sow with cubs or a cub however. I realize that people shoot doe deer with fawns all the time, but the cubs do depend on their mother for the first year of life. I would hold out for a different bear.
ArcticBowMan's Hunting Photo's