  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 $\tau_{erg}$, which itself
is much smaller than the mean dwell time $\tau_{dwell}$. The excitation
spectrum of the Josephson junction has a gap $E_{gap}$, which can be less than
the gap $\Delta$ in the bulk superconductors. The average supercurrent is
computed in the ergodic regime $\tau_{erg} \ll \hbar/\Delta$, using
random-matrix theory, and in the non-ergodic regime $\tau_{erg} \gg
\hbar/\Delta$, 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 $E_{gap}$ and $\Delta$ 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 $\hbar/\tau_{erg}$ greater than $E_{gap} \simeq
\hbar/\tau_{dwell}$. 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.
