Ultraviolet photoelectron spectra (UPS) are reported for mass-selected negative carbon clusters extracted from a pulsed supersonic beam. In the size range from 48 to 84 atoms, three clusters were found to be closed-shell species with appreciable HOMO-LUMO gaps: C50 (0.3-0.6 eV), C60 (1.5-2.0 eV), and C70 (0.7-1.2 eV). UPS data for all other clusters revealed no appreciable HOMO-LUMO gap, indicating they are either open-shell species, or closed-shell species with small HOMO-LUMO gaps. Buckminsterfullerene (C60) was found to have the lowest electron affinity (2.6-2.8 eV) of any cluster. Agreement between these UPS data and electronic structure calculations strongly support the spheroidal shell model for C60.