  This is a study of phase-coherent conduction through a ballistic point
contact with disordered leads. The disorder imposes mesoscopic
(sample-to-sample) fluctuations and weak-localization corrections on the
conductance, and also leads to time-dependent fluctuations (shot noise) of the
current. These effects are computed by means of a mapping onto an unconstricted
conductor with a renormalized mean free path. The mapping holds both in the
metallic and in the localized regime, and permits a solution for arbitrary
ratio of mean free path to sample length. In the case of a single-channel
quantum point contact, the mapping is onto a one-dimensional disordered chain,
for which the complete distribution of the conductance is known. The theory is
supported by numerical simulations. ***Submitted to Physical Review B.****
