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.