If R parity is not an exact symmetry in supersymmetric theories, the lightest supersymmetric partner, most favorably the neutralino, would not be stable, but may decay to produce energetic neutrinos. We show that a lack of excess neutrino events above the atmospheric neutrino background, obtained from upward-going muon data in Kamiokande, places a severe constraint on models of R parity breaking if the neutralino is a dark-matter particle.