The physical characteristics of Pluto and its moon, Charon, provide insight into the evolution of the outer Solar System. Although previous measurements have constrained the masses of these bodies, their radii and densities have remained uncertain. The observation of a stellar occultation by Charon in 1980 established a lower limit on its radius of 600 km (ref. 3) (later refined to 601.5 km; ref. 4) and suggested a possible atmosphere. Subsequent, mutual event modelling yielded a range of 600-650 km (ref. 5), corresponding to a density of 1.56 +/- 0.22 g cm(-3) (refs 2, 5). Here we report multiple-station observations of a stellar occultation by Charon. From these data, we find a mean radius of 606 +/- 8 km, a bulk density of 1.72 +/- 0.15 g cm(-3), and rock-mass fraction 0.63 +/- 0.05. We do not detect a significant atmosphere and place 3sigma upper limits on atmospheric number densities for candidate gases. These results seem to be consistent with collisional formation for the Pluto-Charon system in which the precursor objects may have been differentiated, and they leave open the possibility of atmospheric retention by the largest objects in the outer Solar System.