  The cumulants of the logarithm of the conductance (lng) in the localized
regime in the one-dimensional Anderson model are calculated exactly in the
second Born approximation for weak disorder. Only the first two cumulants turn
out to ne non-zero since the third and fourth cumulants vanish identically and
the higher cumulants are of higher order in the disorder. The variance and the
mean of lng vary linearly with length $L$ while their ratio is proportional to
the inverse localization length. The resulting exact log-normal distribution of
conductance thus corresponds to a special form of two-parameter scaling. This
contradicts the standard one-parameter scaling in the random phase
approximation.
