Can a honey bee be born without the aid of a drone?
Yes and no. A drone's (male bee) purpose is to mate with a queen (female reproductive bee). All other colony activities are performed by worker bees (female bees). To discuss how a bee is born, we can start with when the egg is laid. Generally speaking, if the queen fertilizes this egg with sperm, it will become a worker bee, or another queen. If she does not fertilize the egg, it will become a drone (male). The care and feeding of the larvae that hatches from these eggs are done by worker bees. So you see, in some ways the drone is not required for another drone bee to be born, since sperm is not required for drone bees. Drones are instead required to provide the sperm to fertilize female bees (resulting in genetic recombination), so clearly they are necessary for the species to survive, which is of course required for any honey bee to be born.
-Michael Wilson, University of Tennessee