Due to recent advances in laboratory spectroscopy, the first optical detection of a very large molecule has been claimed in the diffuse interstellar medium (ISM): C60+ (ionized Buckminsterfullerene). Confirming the presence of this molecule would have significant implications regarding the carbon budget and chemical complexity of the ISM. Here we present results from a new method for ultra-high signal-to-noise (S/N) spectroscopy of background stars in the near infrared (at wavelengths 0.9-1 micron), using the Hubble Space Telescope Imaging Spectrograph (STIS) in a previously untested `STIS scan mode. The use of HST provides the crucial benefit of eliminating the need for error-prone telluric correction methods in the part of the spectrum where the C60+ bands lie, and terrestrial water vapor contamination is severe. Our STIS spectrum of the heavily-reddened B0 star BD63,1964 reaches an unprecedented S/N for this instrument ($sim600-800$), allowing the detection of the diffuse interstellar band (DIB) at 9577 AA attributed to C60+ as well as new DIBs in the near-IR. Unfortunately, the presence of overlapping stellar lines, and the unexpected weakness of the C60+ bands in this sightline, prevents conclusive detection of the weaker C60+ bands. A probable correlation between the 9577 AA DIB strength and interstellar radiation field is identified, which suggests that more strongly-irradiated interstellar sightlines will provide the optimal targets for future C60+ searches.
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