There is a supermassive black hole of mass 4 x 10(6) solar masses at the centre of the Milky Way(1,2). A large reservoir of hot (10(7) kelvin) and cooler (10(2) to 10(4) kelvin) gas surrounds it within a few parsecs(3). Although constraints on the amount of hot gas in the accretion zone of the black hole-that is, within 105 Schwarzschild radii (0.04 parsecs)-have been provided by X-ray observations(4-6), the mass in cooler gas has been unconstrained. One possible way this cooler gas could be detected is by its emission in hydrogen recombination spectral lines(7,8). Here we report imaging of a 104-kelvin ionized gas disk within 2 x 10(4) Schwarzschild radii, using the 1.3-millimetre recombination line H30 alpha. This emission line is double-peaked, with full velocity linewidth of about 2,200 kilometres per second. The emission is centred on the radio source Sagittarius A*, but the redshifted side is displaced 0.11 arcsec (0.004 parsecs at a distance of 8 kiloparsecs) to the northeast and the blueshifted side is displaced a similar distance to the southwest. We interpret these observations in terms of a rotating disk of mass 10(-5) to 10(-4) solar masses and mean hydrogen density of about 10(5) to 10(6) per cubic centimetre, with the values being sensitive to the assumed geometry. The emission is stronger than expected, given the upper limit on the strength of the Br gamma spectral line of hydrogen. We suggest that the H30 alpha transition is enhanced by maser emission.