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Single-photon ionization of C60 and C70 has been studied using synchrotron radiation in an energy range from the ionization potential (7.54 +/- 0.04 eV for C60; 7.3 +/-0.2 eV for C70) up to 35 eV. The photoion yield is dominated by a strong resonance at about 20 eV with a FWHM of about 10 eV. This is seen as being due to autoionization via the giant plasmon resonance recently predicted in the RPA approximation. C60 and C70 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. (Author)