We have measured the total cross section for photodetachment of the Na^- ion over the photon energy range 30-51 eV. Electron detachment leads predominantly to the production of Na⁺ ions in this energy range. The structures in the measured cross section are associated with correlated processes involving the detachment or excitation of a 2p core electron, processes which are often accompanied by the excitation of one or more valence electrons. The most prominent feature in the cross section is a strong resonance associated with the excitation of a 2p electron from the core and a 3s valence electron. As in previous experiments on double excitation of valence electrons, electron correlation is seen to play an important role in the dynamics of negative ions.