Air-hydrate crystals store most of the ancient air contained in deep ice sheets. We carried out microscopic observations of air-hydrate crystals below 2000 m depth within the ice core from Dome Fuji, Antarctica, to obtain their number and size distributions. We found that the number density continuously decreased with depth, whereas the average size increased, in contrast to findings from shallower depths. In addition, the characteristic perturbations in both number density and average size distribution with climatic changes almost disappeared, although they are clearly observed in shallow cores. These results indicate that the air-hydrate crystals grow considerably in deeper parts of the ice sheet, and this growth is accompanied by the diffusion of air molecules in the ice. The permeation coefficient of the air molecules in the ice sheet was estimated from the geometric parameters of the air-hydrate distributions. This is the first practical evidence comparable to the previous model estimations. It allows us to evaluate the impacts of the air-molecule migration in the ice sheet on the paleoclimatic information recorded in the deep ice cores.