Abstract
We examine the possibility, remote but not totally improbable, that one of the first two supernova events observed in the Kamiokande II detector consists of an electron neutrino scattering from an electron. From arguments of timing we show that this possibility can be realized only for the first event, and that it requires the electron-neutrino mass to be less than 2.5 eV. The occurrence of such an event means that, of the various Mikheyev-Smirnov-Wolfenstein solutions to the solar-neutrino problem, the nonadiabatic one is likely to be correct.