Single-photon ionization of ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ has been studied using synchrotron radiation in an energy range from the ionization potential (7.54\ifmmode\pm\else\textpm\fi{}0.04 eV for ${\mathrm{C}}_{60}$ 7.3\ifmmode\pm\else\textpm\fi{}0.2 eV for ${\mathrm{C}}_{70}$) up to 35 eV. The photoion yield is dominated by a strong resonance at about 20 eV with a FWHM of \ensuremath{\sim}10 eV. We interpret this as being due to autoionization via the giant plasmon resonance recently predicted in the RPA approximation. ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ offer the unique possibility for a quantitative comparison of this collective excitation in the free molecule with that observed in the solid phase by electron-energy-loss spectroscopy at about 28 eV.