Recent advances in laboratory spectroscopy lead to the claim of ionized Buckminsterfullerene (C-60(+)) as the carrier of two diffuse interstellar bands (DIBs) in the near-infrared. However, irrefutable identification of interstellar C-60(+) requires a match between the wavelengths and the expected strengths of all absorption features detectable in the laboratory and in space. Here we present Hubble Space Telescope (HST) spectra of the region covering the C-60(+) 9348, 9365, 9428, and 9577 angstrom absorption bands toward seven heavily reddened stars. We focus in particular on searching for the weaker laboratory C-60(+) bands, the very presence of which has been a matter for recent debate. Using the novel STIS-scanning technique to obtain ultra-high signal-to-noise spectra without contamination from telluric absorption that afflicted previous ground-based observations, we obtained reliable detections of the (weak) 9365, 9428 angstrom and (strong) 9577 angstrom C-60(+) bands. The band wavelengths and strength ratios are sufficiently similar to those determined in the latest laboratory experiments that we consider this the first robust identification of the 9428 angstrom band, and a conclusive confirmation of interstellar C-60(+).