  The problem of quantum transport in chaotic cavities with broken
time-reversal symmetry is shown to be completely integrable in the universal
limit. This observation is utilised to determine the cumulants and the
distribution function of conductance for a cavity with ideal leads supporting
an arbitrary number $n$ of propagating modes. Expressed in terms of solutions
to the fifth Painlev\'e transcendent and/or the Toda lattice equation, the
conductance distribution is further analysed in the large-$n$ limit that
reveals long exponential tails in the otherwise Gaussian curve.
