Normal incidence x-ray standing wave ͑NIXSW͒ experiments have been performed for monolayers of 3,4,9,10-perylene-tetracarboxylic-dianhydride ͑PTCDA͒ adsorbed on the Ag͑111͒ surface. Two phases were analyzed: the low-temperature phase ͑LT phase͒, which is disordered and obtained for deposition at substrate temperatures below 150 K, and the ordered phase, which is obtained for deposition at room temperature ͑RT phase͒. From the NIXSW analysis the vertical bonding distances to the Ag surface were obtained for the averaged carbon atoms and the two types of chemically different oxygen atoms in the terminal anhydride groups. For the LT phase, we find about 2% ͑0.05 Å͒ and 8% ͑0.21 Å͒ smaller averaged bonding distances for the C and O atoms, respectively, compared to the RT phase. In both phases, the planar geometry of the free molecule is distorted; in particular, the carboxylic O atoms are closer to the surface by 0.20 Å ͑RT͒ and 0.31 Å ͑LT͒ with respect to the averaged C distance. The difference between the vertical bonding distances of the carboxylic and anhydride O atoms is found to be 0.32 ͑RT͒ and 0.33 Å ͑LT͒. These structural parameters of the two phases are compared to those of PTCDA monolayers adsorbed on Au͑111͒ and Cu͑111͒ surfaces and are discussed in the frame of current bonding models.