We consider the distribution function P(|ψ|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 |ψ|2 much larger than the inverse typical localization length ℓ0. Using the recently found solution for the generating function Φan(u, ϕ) we obtain the ALS probability distribution P(|ψ|2) at |ψ|2ℓ0 ≫ 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Φan(u, ϕ) 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 |ψ|2ℓ0, the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of |ψ|2ℓ0, 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(|ψ|2) at small |ψ|2 ≪ ℓ−10 and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.