  Detecting supernova $\nu_e$ is essential for testing supernova and neutrino
physics, but the yields are small and the backgrounds from other channels
large, e.g., $\sim 10^2$ and $\sim 10^4$ events, respectively, in
Super-Kamiokande. We develop a new way to isolate supernova $\nu_e$, using
gadolinium-loaded water Cherenkov detectors. The forward-peaked nature of
$\nu_e + e^- \rightarrow \nu_e + e^-$ allows an angular cut that contains the
majority of events. Even in a narrow cone, near-isotropic inverse beta events,
$\bar{\nu}_e + p \rightarrow e^+ + n$, are a large background. With neutron
detection by radiative capture on gadolinium, the background events can be
individually identified with high efficiency. The remaining backgrounds are
smaller and can be measured separately, so they can be statistically
subtracted. Super-Kamiokande with gadolinium could measure the total and
average energy of supernova $\nu_e$ with $\sim$ $20\%$ precision or better each
($90\%$ C.L.). Hyper-Kamiokande with gadolinium could improve this by a factor
of $\sim$ 5. This precision will allow powerful tests of supernova neutrino
emission, neutrino mixing, and exotic physics. Unless very large liquid argon
or liquid scintillator detectors are built, this is the only way to guarantee
precise measurements of supernova $\nu_e$.

High Energy Astrophysical Phenomena;High Energy Physics - Phenomenology;Nuclear Theory