We use ROSAT PSPC spectra of z approximate to 3 quasars to constrain t
he density and temperature of the intergalactic medium (IGM). Strong l
ow-energy cutoffs in PSPC spectra of high-redshift quasars are common.
However, the absence of absorption toward some high-redshift quasars
can be used to put limits on the possible cosmological density, Omega(
G) of a hot diffuse IGM, via an X-ray ''Gunn-Peterson'' test using edg
e and line opacity in the soft X-rays. The K-edges of oxygen, neon, an
d carbon and the L-edge of iron produce most of the absorption which i
s spread out by the redshift of the source. We assume an isotropic, is
othermal, nonevolving model of the IGM and calculate the optical depth
of this absorption. We find that this test can constrain an enriched
IGM at temperatures near 10(5)-10(6) K, intermediate between the hot I
GM ruled out by COBE, and the cold IGM ruled out by the traditional Ly
alpha Gunn-Peterson test. Photoionization of the IGM by the ultraviol
et and X-ray background has a large effect. We give results for three
z approximate to 3 quasars and discuss how the various trade-offs amon
g temperature, abundance, and background radiation strength affect the
limits on Omega(G). In addition to the high-redshift case, we discuss
techniques for constraining the IGM using X-ray spectra of low-redshi
ft quasars (z approximate to 0.1-0.3). Currently available X-ray spect
ral data have insufficient energy resolution to constrain the IGM unam
biguously, and so expected detection limits for future high-resolution
spectrometers are presented. We find that with a large effective area
(approximate to 2000 cm(2)), it is possible to substantially constrai
n or detect the IGM at the densities which are typically predicted.