  Shot noise in a chaotic cavity (Lyapunov exponent $\lambda$, level spacing
$\delta$, linear dimension $L$), coupled by two $N$-mode point contacts to
electron reservoirs, is studied as a measure of the crossover from stochastic
quantum transport to deterministic classical transport. The transition proceeds
through the formation of {\em fully} transmitted or reflected scattering
states, which we construct explicitly. The fully transmitted states contribute
to the mean current $\bar{I}$, but not to the shot-noise power $S$. We find
that these noiseless transmission channels do not exist for
$N\alt\sqrt{k_{F}L}$, where we expect the random-matrix result
$S/2e\bar{I}=1/4$. For $N\agt\sqrt{k_{F}L}$ we predict a suppression of the
noise $\propto (k_{F}L/N^{2})^{N\delta/\pi\hbar\lambda}$. This nonlinear
contact dependence of the noise could help to distinguish ballistic chaotic
scattering from random impurity scattering in quantum transport.
