We consider binary alloys undergoing phase separation after a deep temperature quench. We present Monte Carlo simulation results showing that such systems freeze when quenched to zero temperature. This holds for a wide range of atomic and vacancy concentrations. We interpret these results in terms of an equilibration mechanism dominated by activated diffusive motion of atoms on a lattice across an interface with an activation energy dependent on the local curvature. We show that in such a case the domain size increases logarithmically for quenches to low temperatures. This logarithmic law is shown to be in excellent agreement with neutron scattering experiments.