We studied exciton structures and the Aharonov-Bohm effect in a single carbon nanotube using micro-photoluminescence (PL) spectroscopy under a magnetic field at low temperatures. A single sharp PL peak from the bright exciton state of a single carbon nanotube was observed under zero magnetic field, and the additional PL of dark exciton state appeared below the bright exciton peak under high magnetic fields. It was found that the split between the bright and dark exciton states is several millielectron volts at zero field. The tube diameter dependence of the splitting arises from the intervalley short-range Coulomb interaction.