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We report a detailed experimental and theoretical study of the electronic structure of \(Mo_{1-x}Re_x\) random alloys. We have measured electronic band dispersions for clean and hydrogen-covered \(Mo_{1-x}Re_x\) ( 110) with x = 0-0.25 using angle-resolved photoemission spectroscopy. Our results suggest that the bulk and most surface electronic bands shift relative to the Fermi level systematically and approximately rigidly with Re concentration. We distinguish and quantify two contributions to these shifts: a raise of the Fermi energy and an increase of the overall bandwidth. Alloy bands calculated using the first-principles Korringa-Kohn-Rostoker coherent-potential-approximation method accurately predict both of these effects. As derived from the rigid band model, the Fermi energy shift is inversely related to the bulk density of states in this energy region. Using our results, we also characterize an electronic topological transition of the bulk Fermi surface and relate this to bulk transport properties. Finally, we distinguish effects beyond the rigid band approximation: a highly surface-localized state and a composition-dependent impact of the spin-orbit interaction.