Using Very Long Baseline Interferometry (VLBI) at the relatively short radio wavelength of 1.3 mm, a new intrinsic size estimate has been obtained for Sagittarius A*, the supermassive black hole candidate at the centre of the Milky Way. The resulting lower limit on the size of Sgr A* is less than the predicted size of the event horizon of the presumed black hole, suggesting that Sgr A* emissions centre not on the black hole itself but on the surrounding accretion flow. VLBI observations of the Galactic Centre at around 1.3 mm, less influenced by interstellar scattering than those made at longer wavelengths, open a new window onto black-hole physics that will become even more sensitive as new VLBI stations are built. The cores of most large galaxies are thought to harbour super massive black holes. Sagittarius A*, the compact source of radio, infrared and x-ray emission at the centre of the Milky Way, is the closest example of this phenomenon. This paper reports observations that set a limit less than the expected apparent size of the event horizon of the presumed black hole, suggesting that the bulk of Sgr A* emission may not be centred on the black hole, but arises in the surrounding accretion flow. The cores of most galaxies are thought to harbour supermassive black holes, which power galactic nuclei by converting the gravitational energy of accreting matter into radiation1. Sagittarius A* (Sgr A*), the compact source of radio, infrared and X-ray emission at the centre of the Milky Way, is the closest example of this phenomenon, with an estimated black hole mass that is 4,000,000 times that of the Sun2,3. A long-standing astronomical goal is to resolve structures in the innermost accretion flow surrounding Sgr A*, where strong gravitational fields will distort the appearance of radiation emitted near the black hole. Radio observations at wavelengths of 3.5 mm and 7 mm have detected intrinsic structure in Sgr A*, but the spatial resolution of observations at these wavelengths is limited by interstellar scattering4,5,6,7. Here we report observations at a wavelength of 1.3 mm that set a size of microarcseconds on the intrinsic diameter of Sgr A*. This is less than the expected apparent size of the event horizon of the presumed black hole, suggesting that the bulk of Sgr A* emission may not be centred on the black hole, but arises in the surrounding accretion flow.