We calculate the cross sections for the neutral-current disintegration of deuterium by neutrinos and antineutrinos: \ensuremath{\nu}+d\ensuremath{\rightarrow}\ensuremath{\nu}'+n+p and \ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{}+d\ensuremath{\rightarrow}\ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{}'+n+p. We put special emphasis on estimates of the theoretical uncertainties of these cross sections. For $^{8}\mathrm{B}$ and hep solar neutrinos, the averaged cross sections are 〈\ensuremath{\sigma}${(}^{8}$B)〉=4.1(1\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}43}$ ${\mathrm{cm}}^{2}$ and 〈\ensuremath{\sigma}(hep)〉=1.15(1\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}42}$ ${\mathrm{cm}}^{2}$, respectively, where hep denotes $^{3}$He+p. The cross-section uncertainty is negligible, \ifmmode\pm\else\textpm\fi{}1/2%, for the ratio of neutral-current to charged-current events. Independent of neutrino oscillations, the cross sections correspond to 4.5(1\ifmmode\pm\else\textpm\fi{}0.38)\ifmmode\times\else\texttimes\fi{}${10}^{3}$ solar-neutrino events per year in the proposed one-kiloton Sudbury Neutrino Observatory if the standard solar model is correct. For a galactic supernova, the total number of neutral-current events expected in the Sudbury detector is about ${10}^{3}$ (distance/8 kpc${)}^{\mathrm{\ensuremath{-}}2}$; most of the signal is expected to arise from \ensuremath{\mu} and \ensuremath{\tau} neutrinos and antineutrinos. If either \ensuremath{\mu} or \ensuremath{\tau} neutrinos have a mass greater than 2\ifmmode\times\else\texttimes\fi{}${10}^{2}$ eV, then this mass should be measurable using the neutrino signal from a galactic supernova.