Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and let $\sigma(E)$ be the self-energy of the model at energy $E$ . For any $\epsilon{>}0$ and sufficiently small $\lambda$ , we derive almost-sure localization in the band $E\le -\sigma(0)-\lambda^{4-\epsilon}$ . In this energy region, we show that the typical correlation length $\xi_E$ behaves roughly as $O\big((|E|-\sigma(E))^{-1/2}\big)$ , completing the argument outlined in the preprint of T. Spencer [18]