The structure of c(2\ifmmode\times\else\texttimes\fi{}2)O on Cu(100) is investigated with a full multiple-scattering analysis of the near-edge x-ray-absorption fine structure (NEXAFS). Various adsorption sites and bond lengths are tested, with the oxygen overlayer found to occupy fourfold hollow sites 0.7 A\r{} above the Cu surface plane, in agreement with recent surface extended x-ray-absorption fine-structure (SEXAFS) work on the same sample. Our study provides the first demonstration that a quantitatively accurate NEXAFS analysis is attainable with low-noise high-resolution data, and opens up new possibilities for surface analysis for those cases where SEXAFS is not applicable.