The structure of the Ag(111)(root 7 x root 7)R19 degrees-CH3S surface phase, formed by interaction of Ag(l 11) with gas-phase dimethyl disulfide, (CH3S)(2), has been investigated by normal-incidence X-ray standing wave (NIXSW) analysis, using (111), ((1) over bar 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).