Cross-section measurements are reported for single and double photoionization of C60+ ions in the photon energy range 18-150 eV accompanied by the loss of zero to seven pairs of carbon atoms, as well as for fragmentation without ionization resulting in loss of two to eight pairs of C atoms in the photon energy range 18-65 eV. Absolute measurements were performed by merging a beam of C60+ molecular ions with a beam of monochromatized synchrotron radiation. Product channels involving dissociation yielding smaller fullerene fragment ions account for nearly half of the total measured oscillator strength in this energy range. The sum of cross sections for the measured product channels is compared to a published calculation of the total photoabsorption cross section of neutral C60 based on time-dependent density-functional theory. This comparison and an accounting of oscillator strengths indicate that with the exception of C58+, the most important product channels resulting from photoabsorption were accounted for in the experiment. Threshold energies for the successive removal of carbon atom pairs accompanying photoionization are also determined from the measurements.