  We consider the distribution function $P(|\psi|^{2})$ of the eigenfunction
amplitude at the center-of-band (E=0) anomaly in the one-dimensional
tight-binding chain with weak uncorrelated on-site disorder (the
one-dimensional Anderson model). The special emphasis is on the probability of
the anomalously localized states (ALS) with $|\psi|^{2}$ much larger than the
inverse typical localization length $\ell_{0}$. Using the solution to the
generating function $\Phi_{an}(u,\phi)$ found recently in our works we find the
ALS probability distribution $P(|\psi|^{2})$ at $|\psi|^{2}\ell_{0} >> 1$. As
an auxiliary preliminary step we found the asymptotic form of the generating
function $\Phi_{an}(u,\phi)$ at $u >> 1$ which can be used to compute other
statistical properties at the center-of-band anomaly. We show that at
moderately large values of $|\psi|^{2}\ell_{0}$, the probability of ALS at E=0
is smaller than at energies away from the anomaly. However, at very large
values of $|\psi|^{2}\ell_{0}$, the tendency is inverted: it is exponentially
easier to create a very strongly localized state at E=0 than at energies away
from the anomaly. We also found the leading term in the behavior of
$P(|\psi|^{2})$ at small $|\psi|^{2}<< \ell_{0}^{-1}$ and show that it is
consistent with the exponential localization corresponding to the Lyapunov
exponent found earlier by Kappus and Wegner and Derrida and Gardner.
