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The structures of ordered phases following adsorption of mercury on Cu(001) have been identified using helium-beam scattering and low-energy electron diffraction. We found that three of the four observed ordered submonolayer phases are high-order commensurate phases. The three-dimensional structure of these phases brought about by a periodic variation of the height of the Hg atoms above the Cu(001) plane has been measured. Furthermore, from a combined analysis of helium-beam scattering and angle-resolved photoemission data for a low-density Hg ordered overlayer, we deduce and quantify the departure from the atomic electron distribution of Hg due to adsorption. A model to analyze diffraction data from all these phases is also presented. We show that second-layer structural and electronic properties are influenced by those of the first layer. Adsorption at the lowest substrate temperature obtainable proceeds in a layer-by-layer mode up to three layers.