The adsorption geometry of perfluorinated copper-phthalocyanine molecules (F_{16}CuPc) on Cu(111) and Ag(111) is studied using X-ray standing waves. A detailed, element-specific analysis taking into account non-dipolar corrections to the photoelectron yield shows that on both surfaces the molecules adsorb in a lying down, but significantly distorted configuration. While on copper (silver) the central carbon rings reside 2.61 Angstroem (3.25 Angstroem) above the substrate, the outer fluorine atoms are located 0.27 Angstroem (0.20 Angstroem) further away from the surface. This non-planar adsorption structure is discussed in terms of the outer carbon atoms in F_{16}CuPc undergoing a partial rehybridization (sp^2 --> sp^3). Comment: 8 pages, 7 figures, for further information see http://www.physchem.ox.ac.uk/~fs