  An exact solution is found for the problem of the center-of-band ($E=0$)
anomaly in the one-dimensional (1D) Anderson model of localization. By deriving
and solving an equation for the generating function $\Phi(u,\phi)$ we obtained
an exact expression in quadratures for statistical moments $I_{q}=\langle
|\psi_{E}({\bf r})|^{2q}\rangle$ of normalized wavefunctions $\psi_{E}({\bf
r})$ which show violation of one-parameter scaling and emergence of an
additional length scale at $E\approx 0$.
