If the neutrino mass is nonzero, as hinted by several experiments, then R-parity-violating supersymmetric Yukawa couplings can drive a heavy neutrino decay into lighter states. The heavy neutrino may either decay radiatively into a lighter neutrino, or it may decay into three light neutrinos through a Z-mediated penguin diagram. For a given mass of the decaying neutrino, we calculate its lifetime for the various modes, each mode requiring certain pairs of R-parity-violating couplings to be nonzero. We then check whether the calculated lifetimes fall in zones allowed or excluded by cosmological requirements. For the latter case, we derive stringent new constraints on the corresponding products of R-parity-violating couplings for given values of the decaying neutrino mass. ©2000 The American Physical Society.