One of the most important considerations when planning the next generation of ground-based optical astronomical telescopes is to choose a site that has excellent 'seeing'--the jitter in the apparent position of a star that is caused by light bending as it passes through regions of differing refractive index in the Earth's atmosphere. The best mid-latitude sites have a median seeing ranging from 0.5 to 1.0 arcsec (refs 1-5). Sites on the Antarctic plateau have unique atmospheric properties that make them worth investigating as potential observatory locations. Previous testing at the US Amundsen-Scott South Pole Station has, however, demonstrated poor seeing, averaging 1.8 arcsec (refs 6, 7). Here we report observations of the wintertime seeing from Dome C (ref. 8), a high point on the Antarctic plateau at a latitude of 75 degrees S. The results are remarkable: the median seeing is 0.27 arcsec, and below 0.15 arcsec 25 per cent of the time. A telescope placed at Dome C would compete with one that is 2 to 3 times larger at the best mid-latitude observatories, and an interferometer based at this site could work on projects that would otherwise require a space mission.