Atomically thin hexagonal boron nitride (h-BN) layers on metallic supports represent a promising platform for the selective adsorption of atoms, clusters, and molecular nanostructures. Specifically, scanning tunneling microscopy (STM) studies revealed an electronic corrugation of h-BN/Cu(111), guiding the self-assembly of molecules and their energy level alignment. A detailed characterization of the h-BN/Cu(111) interface including the spacing between the h-BN sheet and its support-elusive to STM measurements-is crucial to rationalize the interfacial interactions within these systems. To this end, we employ complementary techniques including high-resolution noncontact atomic force microscopy, STM, low-energy electron diffraction, X-ray photoelectron spectroscopy, the X-ray standing wave method, and density functional theory. Our multimethod study yields a comprehensive, quantitative structure determination including the adsorption height and the corrugation of the sp2 bonded h-BN layer on Cu(111). Based on the atomic contrast in atomic force microscopy measurements, we derive a measurable-hitherto unrecognized-geometric corrugation of the h-BN monolayer. This experimental approach allows us to spatially resolve minute height variations in low-dimensional nanostructures, thus providing a benchmark for theoretical modeling. Regarding potential applications, e.g., as a template or catalytically active support, the recognition of h-BN on Cu(111) as a weakly bonded and moderately corrugated overlayer is highly relevant.