  We present results from an extensive HST spectroscopic search for UV
absorption lines in the spectrum of the quasar B2~1028+313, which is associated
with the central dominant galaxy in the cluster Abell~1030 ($z=0.178$). This is
one of the brightest known UV continuum sources located in a cluster, and
therefore provides an ideal opportunity to obtain stringent constraints on the
column densities of any cool absorbing gas that may be associated with the
intracluster medium (ICM). Our HST spectra were obtained with the FOS and GHRS,
and provide continuous coverage at rest-frame wavelengths from $\sim 975$ to
4060~\AA, thereby allowing the investigation of many different elements and
ionization levels. We utilize a new technique that involves simultaneous
fitting of large numbers of different transitions for each species, thereby
yielding more robust constraints on column densities than can be obtained from
a single transition. This method yields upper limits of $\lesssim 10^{11} -
10^{13}$ cm$^{-2}$ on the column densities of a wide range of molecular, atomic
and ionized species that may be associated with the ICM. We also discuss a
possible \Lya and C IV absorption system associated with the quasar. We discuss
the implications of the upper limits on cool intracluster gas in the context of
the physical properties of the ICM and its relationship to the quasar.
