  A closed expression is derived for the probability distribution of the
transfer matrix of a particle moving in a one-dimensional system with
delta-correlated, weak disorder. The change in the distribution as a function
of wire length is described by a diffusion equation on the $SU(1,1)$ group,
which is solved through the decomposition of the regular representation into
irreducible components. The expression generalizes a number of well-known
results, including the distributions of the transmission coefficient and local
density of states. As an application, the average single energy-level
contribution to the persistent current in a flux-threaded ring is derived.
