Laboratory measurements are reported that enable a more accurate determination of the characteristics of the near-infrared absorptions of C-60(+) below 10 K. These data were obtained by photofragmentation of C-60(+)-He complexes in a cryogenic trap. Asymmetry in the profiles of the observed 9577 and 9632 angstrom absorption bands of C-60(+)-He is caused by the attachment of the weakly bound helium atom to hexagonal or pentagonal faces of C-60(+). The implication is that the FWHM of the bands in the electronic spectrum of C-60(+) below 10 K is 1.4 angstrom. The effect of C-13 isotopes on the C-60(+) electronic spectrum is experimentally evaluated by measurement of C-12(60)+-He, (C1C59+)-C-13-C-12-He, and (C2C58+)-C-13-C-12-He. Data on the 9365 angstrom absorption band indicate a wavelength shift of about 0.3 angstrom between the former and latter. This result is consistent with models used to interpret the vibrational isotope effect in the Raman spectrum of neutral C-60. The influence of C-13 isotopes on the 9348, 9365, 9428, 9577, and 9632 angstrom diffuse interstellar bands is expected to be minor considering other broadening factors that affect astronomical observations. The presented data also provide more accurate relative intensities of the five interstellar bands attributed to C-60(+).