Molecular diffusion in amorphous solid water is studied by measuring the extent of isotope exchange and mixing in vapor-deposited ultrathin films using temperature-programmed desorption. Our results indicate there is long-range translational diffusion that occurs concomitantly with the amorphous to crystalline ice phase transition. The apparent diffusivity estimated from these data is a factor of 10(6+/-1) greater than an Arrhenius extrapolation for diffusion in crystalline ice. This finding suggests that the amorphous material exhibits liquid-like translational diffusion prior to crystallization at temperatures near 155 K.