  Near-infrared magneto-optical spectroscopy of single-walled carbon nanotubes
reveals two absorption peaks with an equal strength at high magnetic fields
($>$ 55 T). We show that the peak separation is determined by the Aharonov-Bohm
phase due to the tube-threading magnetic flux, which breaks the time-reversal
symmetry and lifts the valley degeneracy. This field-induced symmetry breaking
thus overcomes the Coulomb-induced intervalley mixing which is predicted to
make the lowest exciton state optically inactive (or ``dark'').
