  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 Songaila's (1998) report that the
hardness of the UV background spectrum increases rapidly from z=3 to z=2.9.
