We calculate the supercurrent through a Josephson junction consisting of a phase-coherent metal particle (quantum dot), weakly coupled to two superconductors. The classical motion in the quantum dot is assumed to be chaotic on time scales greater than the ergodic time r,,s, which itself is much smaller than the mean dwell time r dWe,,. The excitation spectrum of the Josephson junction has a gap E,,,, which can be less than the gap A in the bulk superconductors. The average supercurrent is computed in the ergodic regime r,, <> fi/A, using a semiclassical relation between the supercurrent and dwell-time distribution. In contrast to conventional Josephson junctions, raising the temperature above the excitation gap does not necessarily lead to an exponential suppression of the supercurrent. Instead, we find a temperature regime between E,, and A where the supercurrent decreases logarithmically with temperature. This anomalously weak temperature dependence is caused by long-range correlations in the excitation spectrum, which extend over an energy range fi/r,,, greater than E,,, = ~5/7~~~,,. A similar logarithmic temperature dependence of the supercurrent was discovered by Aslamazov, Larkin and Ovchinnikov in a Josephson junction consisting of a disordered metal between two tunnel barriers.