Accretion of matter onto the surface of a white dwarf in a binary system can push it over the Chandrasekhar mass limit and may cause it to collapse into a naked or nearly naked neutron star without detectable optical emission. Such ali optically quiet neutron star birth should be accompanied by a neutrino burst which could be detected with underground neutrino detectors only if the collapse took place in our own Galaxy or in very close nearby galaxies. However, neutrino-antineutrino annihilation outside the neutron star into electron-positron pairs will produce a gamma-ray burst that can be observed out to distances of at least 300 Mpc, if the mass surrounding the newly formed neutron star is less than about 3 x 10(-4) M.. If the surrounding mass is between approximately 3 x 10(-4) M. and approximately 0.1 M., it will be injected into the interstellar space with energy above 10 meV per nucleon. Such nonrelativistic nuclei can be further accelerated to cosmic-ray energies in the interstellar space before they slow down by collisions. Thus, accretion-induced collapse may be an important source of cosmic rays and of cosmological gamma-ray bursts. Conversely, the observed rate of gamma-ray bursts and cosmic-ray data can be used to limit the birthrate of naked, or nearly naked, neutron stars to less than one per 10(3) yr in galaxies similar to ours. This rate is too small to contribute significantly to the birthrate of pulsars, and it implies that it is very unlikely that a neutrino burst unaccompanied by optical emission will be detected in the near future by the underground neutrino detectors.