  The $\nu_\mu$ and $\nu_\tau$ neutrinos (and their antiparticles) from a
Galactic core-collapse supernova can be observed in a water-\v{C}erenkov
detector by the neutral-current excitation of $^{16}$O. The number of events
expected is several times greater than from neutral-current scattering on
electrons. The observation of this signal would be a strong test that these
neutrinos are produced in core-collapse supernovae, and with the right
characteristics. In this paper, this signal is used as the basis for a
technique of neutrino mass determination from a future Galactic supernova. The
masses of the $\nu_\mu$ and $\nu_\tau$ neutrinos can either be measured or
limited by their delay relative to the $\bar{\nu}_e$ neutrinos. By comparing to
the high-statistics $\bar{\nu}_e$ data instead of the theoretical expectation,
much of the model dependence is canceled. Numerical results are presented for a
future supernova at 10 kpc as seen in the SuperKamiokande detector. Under
reasonable assumptions, and in the presence of the expected counting
statistics, $\nu_\mu$ and $\nu_\tau$ masses down to about 50 eV can be simply
and robustly determined. The signal used here is more sensitive to small
neutrino masses than the signal based on neutrino-electron scattering.
