The radio source Sagittarius A^* (Sgr A^*), is thought to be a supermassive black hole located at the centre of our Galaxy, 1,2 that is accreting gas from the surrounding region. Using the high inferred accretion rates,3 however, standard accretion models4 are unable to explain the low luminosity and observed spectrum from Sgr A*.5-8 A new accretion model has been proposed - an advection dominated accretion flow (ADAF),9-12 where most of the accretion energy is stored in the gas and lost into the black hole. The gas therefore has a two-temperature structure10, 13, 14 with the protons being much hotter than the electrons. The model explains the low luminosity from Sgr A* 15-18 and most of the millimeter to hard X-ray spectrum, but has had serious difficulty in agreeing with the low energy radio observations.18 Here we report an emission process associated with the protons that naturally resolves the observed discrepancy. This provides, for the first time, observational evidence for a two-temperature plasma in hot accretion flows, and gives strong support to the idea that an ADAF does accrete onto a 2.5 * 106 solar mass black hole at the Galactic Centre.