First results of numerical simulations are presented which compute the dynamical evolution of a neutron star with a mass slightly below the minimum stable mass by means of a new implicit (general relativistic) hydrodynamic code. We show that such a star first undergoes a phase of quasi-static expansion, caused by slow nuclear beta -decays, lasting for about 20 seconds, but then explodes violently. The kinetic energy of the explosion is around 10(49) erg, the peak luminosity in electron anti-neutrinos is of order 10(52) erg/s, and the thermodynamic conditions of the expanding matter are favorable for r-process nucleosynthesis. These results are obtained for the Harrison-Wheeler equation of state and a simple and, possibly, unrealistic treatment of beta -decay rates and nuclear fission, which were adopted for comparison with previous works. However, we do not expect that the outcome will change qualitatively if more recent nuclear input physics is used. Although our study does not rely on a specific scenario of how a neutron star starting from a bigger (and stable) mass can reach the dynamical phase, we implicitly assume that the final mass-loss event happens on a very short time scale, i.e., on a time scale shorter than a sound-crossing time, by removing a certain amount of mass as an initial perturbation. This assumption implies that the star has no time to adjust its nuclear composition to the new mass through a sequence of quasi-equilibria. In the latter case, however, there exists no stable configuration below the minimum mass, because the equation of state of fully catalyzed matter is too soft. Therefore, the dynamics of the explosion will not be too different from what we have obtained if different initial perturbations are assumed.