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Seeing, the angular size of stellar images blurred by atmospheric turbulence, is a critical parameter used to assess the quality of astronomical sites. Median values at the best mid-latitude sites are generally in the range of 0.6--0.8\,arcsec. Sites on the Antarctic plateau are characterized by comparatively-weak turbulence in the free-atmosphere above a strong but thin boundary layer. The median seeing at Dome C is estimated to be 0.23--0.36 arcsec above a boundary layer that has a typical height of 30\,m. At Dome A and F, the only previous seeing measurements were made during daytime. Here we report the first direct measurements of night-time seeing at Dome A, using a Differential Image Motion Monitor. Located at a height of just 8\,m, it recorded seeing as low as 0.13\,arcsec, and provided seeing statistics that are comparable to those for a 20\,m height at Dome C. It indicates that the boundary layer was below 8\,m 31\% of the time. At such times the median seeing was 0.31\,arcsec, consistent with free-atmosphere seeing. The seeing and boundary layer thickness are found to be strongly correlated with the near-surface temperature gradient. The correlation confirms a median thickness of approximately 14\,m for the boundary layer at Dome A, as found from a sonic radar. The thinner boundary layer makes it less challenging to locate a telescope above it, thereby giving greater access to the free-atmosphere.