Early black hole formation in core-collapse supernovae may be triggered by mass accretion or a change in the high-density equation of state. We consider the possibility that black hole formation happens when the flux of neutrinos is still measurably high. If this occurs, then the neutrino signal from the supernova will be terminated abruptly (the transition takes $lesssim 0.5$ ms). The properties and duration of the signal before the cutoff are important measures of both the physics and astrophysics of the cooling proto-neutron star. For the event rates expected in present and proposed detectors, the cutoff will generally appear sharp, thus allowing model-independent time-of-flight mass tests for the neutrinos after the cutoff. If black hole formation occurs relatively early, within a few ($sim 1$) seconds after core collapse, then the expected luminosities are of order $L_{BH} = 10^{52}$ erg/s per flavor. In this case, the neutrino mass sensitivity can be extraordinary. For a supernova at a distance $D = 10$ kpc, SuperKamiokande can detect a $bar{ u}_e$ mass down to 1.8 eV by comparing the arrival times of the high-energy and low-energy neutrinos in $bar{ u}_e + p to e^+ + n$. This test will also measure the cutoff time, and will thus allow a mass test of $ u_mu$ and $ u_tau$ relative to $bar{ u}_e$. Assuming that $ u_mu$ and $ u_tau$ are nearly degenerate, as suggested by the atmospheric neutrino results, masses down to about 6 eV can be probed with a proposed lead detector of mass $M_D = 4$ kton (OMNIS). Remarkably, the neutrino mass sensitivity scales as $(D/L_{BH} M_D)^{1/2}$. Therefore, {it direct} sensitivity to all three neutrino masses in the interesting few-eV range is realistically possible; {it there are no other known techniques that have this capability}.
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