We present the first complete LEED-IV analysis of an ordered water layer adsorbed on a metal surface, the structure of p(square-root 3 x square-root 3)R30-degrees-2D2O/Ru(001), commonly assumed to be a bilayer. The best fit geometry is a buckled hexagonal arrangement of water molecules adsorbed on top of first-layer ruthenium atoms. The main, at first sight surprising feature of the surface geometry is that the water ''bilayer'' is almost flat: the vertical distance between the O atoms of the two types of molecules in the p(square-root 3 x square-root 3)R30-degrees unit mesh (DELTAz(O-O) is 0.10 +/- 0.02 angstrom. Buckling of the first Ru layer is found in anticorrelation to the water molecules (0.01 +/- 0.04 angstrom and 0.07 +/- 0.02 angstrom with respect to the uncovered Ru atoms), so that the Ru-O distances for the two types of molecules are still significantly different from each other (2.08 +/- 0.02 angstrom and 2.23 +/- 0.02 angstrom). We conclude from this that the construction principle of the bilayer (chemical bond between the Ru atoms and the O atoms of the closer molecules, and hydrogen bonding from the latter to the other molecules) is still correct. Dipole-mirror dipole attraction and charge-transfer-induced changes of the D2O bond angle are discussed as possible explanations for the compression of the ideal bilayer structure, which cannot be explained by the influence of the lattice mismatch between Ru(001) and ice. The vertical distance between the first- and second-layer Ru atoms ranges from 2.04 +/- 0.04 angstrom to 2.11 +/- 0.03 angstrom due to the first-layer buckling, i.e. is always - and in part considerably - smaller than in the bulk.