THE radio source at the centre of our Galaxy(1), Sagittarius A* (Sgr A*), seems to be a low-luminosity version of active galactic nuclei-a massive black hole that is accreting gas from the surrounding region(1,2). This idea is supported by observations of the gas and stars within 1 pc of Sgr A*, which appear to move under the influence of a large central mass(1,3,4). A recent determination of the upper limit(5,6) to the hard X-ray emission from the Galactic Centre has posed a problem for this picture, however, as the mass accretion rate implied by applying a standard accretion model to the X-ray data is far below that estimated from the observations of gas flows. Here we present a new model of accretion onto Sgr A*, in which most of the energy released is carried along with the gas and lost into the black hole of mass similar to 7 x 10(5) solar masses, rather than appearing as radiation. The model fits the observed spectrum of Sgr A* from radio to hard X-ray wavelengths, and reconciles the low observed luminosity with a high mass-accretion rate.