A numerical procedure for the integration of the reactor kinetics equation is developed. It has the property of being numerically unconditionally stable for all values of the reactivity or integration-step size. The basic assumption of the method is that the neutron and precursor densities behave exponentially with a frequency characteristic of the asymptotic frequency corresponding to the reactivity. As a consequence of the assumption, and the factoring of the kinetics equation, it is then shown that for constant reactivity the asymptotic numerical eigensolution is exactly the asymptotic eigensolution of the differential kinetics equations. Thus, for constant reactivity, the asymptotic numerical solution can be shown to differ from the asymptotic analytic solution by at most a constant factor, proportional to ht2, for all time.