Neutrino-electron elastic scattering was observed with a 15-ton fine-grained tracking calorimeter exposed to electron neutrinos from muon decay at rest. The measured ${\ensuremath{\nu}}_{e}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\nu}}_{e}{e}^{\ensuremath{-}}$ elastic scattering rate of 236\ifmmode\pm\else\textpm\fi{}35 events yields the total elastic scattering cross section $10.0\ifmmode\pm\else\textpm\fi{}1.5(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.9(\mathrm{syst})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}45} \mathrm{c}{\mathrm{m}}^{2}\ifmmode\times\else\texttimes\fi{}[{E}_{\ensuremath{\nu}} (\mathrm{MeV})]$, and a model-independent measurement of the strength of the destructive interference between the charged and neutral currents, $I=\ensuremath{-}1.07\ifmmode\pm\else\textpm\fi{}0.21$, that agrees well with the standard model (SM) prediction $I=\ensuremath{-}1.08$. The agreement between the measured electroweak parameters and SM expectations is used to place limits on neutrino properties, such as neutrino flavor-changing neutral currents and neutrino electromagnetic moments. Limits are placed on the masses of new bosons that interact with leptons: for a neutral tensor boson, ${M}_{T}g105$ GeV; for a neutral (pseudo)scalar boson, ${M}_{P,S}g47$ GeV; for a charged Higgs boson, ${M}_{{\ensuremath{\chi}}^{+}}g87$ GeV; and for a purely left-handed charged (neutral) vector boson, ${M}_{x}g239 (119)$ GeV.