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We report the measurement, using angle-resolved photoemission spectroscopy, of the metallic electronic structure of the hole-doped thermoelectric oxide CuRh$_{0.9}$Mg$_{0.1}$O$_2$. The material is found to have a ``pudding mold'' type band structure, with a nearly flat band edge located near the Fermi level, which is thought to be the origin of the thermoelectric behavior of this material. The experimental data match the density functional theory of the undoped parent compound, simply corrected by a rigid shift of the bands. Transport calculations based on the observed band structure yield a Seebeck coefficient of $\sim 200 \,\mu$V/K for the undoped parent material, consistent with experimental measurements. Our results show that CuRhO$_2$ is a textbook example of how pure band-structural effects can result in a large thermoelectric figure of merit, demonstrating that flat band edges in oxides are a realistic route for the efficient conversion of thermal energy.