At low temperatures, liquids imbibed into nanoporous materials form frozen solid cores in the interior of pores. Inbetween the cores and the pore walls, there exists a layer of nonfrozen liquid. As demonstrated here by nuclear magnetic resonance magnetization transfer and cryoporometry experiments, pools of nuclear spins that reside in those two separate domains, core and liquid layer, exchange nuclear Zeeman magnetization. This exchange is detectable in these materials thanks to the high contact area. Based on this phenomenon, we propose a new method for measuring the surface-to-volume ratio and for accessing the pore space morphology in nanoporous materials. By monitoring the magnetization transfer process, information can be obtained: (i) on the geometrical characteristics of the porous space and (ii) on the diffusivity of nuclear magnetization in the frozen phase. The diffusive transport of nuclear magnetization in the frozen core can be contributed by two mechanisms: spatial atomic/molecular diffusion and spin diffusion through nuclear dipole-dipole interaction. For the water-ice system in controlled porous glasses of known morphology, we can separate these two mechanisms to obtain an estimate of the proton diffusion coefficient of 4x10(-16) m(2)/s in ice at 255 K.