We present a model where an unstable MeV Majorana tau neutrino can naturally reconcile the cold dark matter model (CDM) with cosmological observations of large and small scale density fluctuations and, simultaneously, with data on solar and atmospheric neutrinos. The solar neutrino deficit is explained through long wavelength, so-called just-so oscillations involving conversions of nu(e) into both nu(mu) and a sterile species nu(s), while atmospheric neutrino data are explained through nu(mu) to nu(e) conversions. Future long baseline neutrino oscillation experiments, as well as some reactor experiments will test this hypothesis. The model is based on the spontaneous violation of a global lepton number symmetry at the weak scale. This symmetry plays a key role in generating the cosmologically required decay of the nu(tau) with lifetime tau(nu tau) similar to 10(2)-10(4) seconds, as well as the masses and oscillations of the three light neutrinos nu(e), nu(mu), and nu(s) required in order to account for solar and atmospheric neutrino data. It also leads to the invisibly decaying Higgs signature that can be searched at LEP and future particle colliders.