The one-dimensional spin-exchange kinetic Ising model is studied using approximations based on the motion of single spins. This model exhibits domain-scaling behavior after a deep quench to low temperatures, with the same scaling exponent (1/3) as in higher dimensions. Under slow cooling, the kink density of this system is predicted to freeze at a value proportional to τ−1/z, where τ is the inverse cooling rate and z is the dynamic critical exponent (=5) for ‘‘natural’’ cooling programs. The results of Monte Carlo simulations are found to compare favorably with these predictions. The residual temporal behavior in a frozen nonequilibrium state is studied in the short- and long-time regimes, approaching asymptotically a stretched-exponential form.