The structure of the Ag(111)(radical7x radical7)R19 degrees-CH3S surface phase, formed by interaction of Ag(111) with gas-phase dimethyl disulfide, (CH3S)2, has been investigated by normal-incidence X-ray standing wave (NIXSW) analysis, using (111), (11), and (200) Bragg reflections. The resulting NIXSW structural parameter values clearly exclude any simple overlayer adsorption model on an Ag unreconstructed surface. A reconstructed surface model is proposed that is consistent with the NIXSW measurements and with previous scanning tunneling microscopy results. This comprises a near-hexagonal Ag surface layer with an Ag density of only 3/7 that of the underlying substrate layers; the methanethiolate molecules are adsorbed into 3-fold coordinated hollow sites on this open layer. The results are discussed in the context of the very limited published studies of longer alkyl chain thiolates on Ag(111).