  We investigate the distribution of the supercurrent through a chaotic quantum
dot which is strongly coupled to two superconductors when the Thouless energy
is large compared to the superconducting energy gap. The distribution function
of the critical currents P(I_c) is known to be Gaussian in the limit of large
channel number N. For N=1, we present an analytical low-temperature expression
for this distribution function, valid both in the presence and in the absence
of time-reversal symmetry. It connects directly the distribution of
transmission coefficients to the distribution of critical currents. The case of
arbitrary channel number (N>1) is discussed numerically, and for small critical
currents analytically.
