The structure of the oxygen-induced p(4 x 4) reconstruction of Ag(111) is determined by a combination of scanning tunneling microscopy, surface x-ray diffraction, core level spectroscopy, and density functional theory. We demonstrate that all previous models of this surface structure are incorrect and propose a new model which is able to explain all our experimental findings but has no resemblance to bulk silver oxide. We also shed some light on the limitations of current density functional theories and the potential role of van der Waals interactions in the stabilization of oxygen-induced surface reconstructions of noble metals.