We used angle-resolved photoemission spectroscopy to measure the electronic dispersion and single-particle spectral function in a liquid metal. A lead monolayer supported on a copper (111) surface was investigated as the temperature was raised through the melting transition of the film. Electron spectra and momentum distribution maps of the liquid film revealed three key features of the electronic structure of liquids: the persistence of a Fermi surface, the filling of band gaps, and the localization of the wave functions upon melting. Distinct coherence lengths for different sheets of the Fermi surface were found, indicating a strong dependence of the localization lengths on the character of the constituent atomic wave functions.