We study a class of nonlinear Schroedinger equations admitting small solitary wave solutions. If the linearized equation has a single bound state, we prove that all solutions small in the energy space H1 split, asymptotically, into a fixed nonlinear ground state and a solution of the free Schr¨odinger equation. If the linearized operator has a second ”wellplaced” bound state, we prove the same thing for data which is an H1-small perturbation of the nonlinear ground state. In particular, we do not make the usual ”localization” assumption on the initial data.