The diffuse interstellar bands are absorption lines seen towards reddened stars. None of the molecules responsible for these bands have been conclusively identified. Two bands at 9,632 ångströms and 9,577 ångströms were reported in 1994, and were suggested to arise from C60+ molecules (ref. 3), on the basis of the proximity of these wavelengths to the absorption bands of C60+ measured in a neon matrix. Confirmation of this assignment requires the gas-phase spectrum of C60+. Here we report laboratory spectroscopy of C60+ in the gas phase, cooled to 5.8 kelvin. The absorption spectrum has maxima at 9,632.7 +/- 0.1 ångströms and 9,577.5 +/- 0.1 ångströms, and the full widths at half-maximum of these bands are 2.2 +/- 0.2 ångströms and 2.5 +/- 0.2 ångströms, respectively. We conclude that we have positively identified the diffuse interstellar bands at 9,632 ångströms and 9,577 ångströms as arising from C60+ in the interstellar medium.