The Trojan population consists of two swarms of asteroids following the same orbit as Jupiter and located at the L4 and L5 stable Lagrange points of the Jupiter-Sun system (leading and following Jupiter by 60°). The asteroid 617Patroclus is the only known binary Trojan. The orbit of this double system was hitherto unknown. Here we report that the components, separated by 680km, move around the system's centre of mass, describing a roughly circular orbit. Using this orbital information, combined with thermal measurements to estimate the size of the components, we derive a very low density of 0.8 - 0.1 + 0.2gcm-3. The components of 617Patroclus are therefore very porous or composed mostly of water ice, suggesting that they could have been formed in the outer part of the Solar System.