The unique physical properties of buckminsterfullerene, C-60, have attracted intense research activity since its original discovery. Total quantum state-resolved spectroscopy of isolated C-60 molecules has been of particularly long-standing interest. Such observations have, to date, been unsuccessful owing to the difficulty in preparing cold, gas-phase C-60 in sufficiently high densities. Here we report high-resolution infrared absorption spectroscopy of C-60 in the 8.5-micron spectral region (1180 to 1190 wave number). A combination of cryogenic buffer-gas cooling and cavity-enhanced direct frequency comb spectroscopy has enabled the observation of quantum state-resolved rovibrational transitions. Characteristic nuclear spin statistical intensity patterns confirm the indistinguishability of the 60 carbon-12 atoms, while rovibrational fine structure encodes further details of the molecule's rare icosahedral symmetry.