The ultraviolet spectrum (1145--1720A) of the distant quasar Q 0302--003 (z=3.286) was observed at 1.8A resolution with the Space Telescope Imaging Spectrograph aboard the Hubble Space Telescope. A total integration time of 23,280 s was obtained. The spectrum clearly delineates the Gunn-Peterson HeII absorption trough, produced by HeII Lya, along the line of sight over the redshift range z=2.78-3.28. Its interpretation was facilitated by modeling based on Keck HIRES spectra of the HI Lya forest (provided by A. Songaila and by M. Rauch and W. Sargent). We find that near the quasar HeII Lya absorption is produced by discrete clouds, with no significant diffuse gas; this is attributed to a HeII proximity effect in which the quasar fully ionizes He in the diffuse intergalactic medium, but not the He in denser clouds. By two different methods we calculate that the average HeII Lya opacity at z~3.15 is tau >= 4.8. In the Dobrzycki-Bechtold void in the HI Lya forest near z=3.18, the average HeII opacity tau=4.47^{+0.48}_{-0.33}. Such large opacities require the presence of a diffuse gas component as well as a soft UV background spectrum, whose softness parameter, defined as the ratio of the photo-ionization rate in HI over the one in HeII, S=Gamma^J_HI/Gamma^J_HeII~=800, indicating a significant stellar contribution. At z=3.05, there is a distinct region of high HeII Lya transmission which most likely arises in a region where helium is doubly ionized by a discrete local source, quite possibly an AGN. At redshifts z<2.87, the HeII Lya opacity detected by STIS, tau=1.88, is significantly lower than at z>3. Such a reduction in opacity is consistent with Songailas (1998) report that the hardness of the UV background spectrum increases rapidly from z=3 to z=2.9.
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