We observed several nearby face-on spiral galaxies with the ROSAT PSPC. The apparent deficiency in soft X-ray surface brightness observed at the outer portion of their disks is consistent with the absorption of the extragalactic soft X-ray background by material associated with these galaxies, and allows us to place a lower limit on the intensity of this cosmologically important background. From the depth of the soft X-ray shadow observed in NGC 3184, a 95% confidence lower limit was derived to be $32 keV cm^{-2} s^{-1} keV^{-1}$ at 1/4 keV. This was obtained by assuming that there is no unresolved 1/4 keV X-ray emission from the outer region of the galaxy which may otherwise partially fill in the shadow: any such emission, or any unresolved structure in the absorbing gas, would imply a larger value. In the deepest exposure to date in this energy range, Hasinger et al. (1993) resolved about $30 keV cm^{-2} s^{-1} keV^{-1}$ at 1/4 keV into discrete sources; our current limit is therefore consistent with an extragalactic origin for all of these sources. Our results can also be directly compared with the corresponding upper limit derived from the ROSAT PSPC detection of soft X-ray shadows cast by high-latitude clouds in Ursa Major, $simeq 65 keV cm^{-2} s^{-1} keV^{-1}$ at 1/4 keV. The lower and upper limits are only a factor of 2 apart, and begin to provide a reasonable measurement of the intensity of the 1/4 keV extragalactic X-ray background.
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