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We have used the Hubble Space Telescope (HST) Goddard High Resolution Spectrograph (GHRS) to search for Ly$\alpha$ absorption clouds in nearby galaxy voids ($cz \leq 10,000$ km~s$^{-1}$). Thus far, we have obtained GHRS spectra (G160M, 1225 -- 1255 \AA, 0.25 \AA\ resolution) of three very bright Active Galactic Nuclei, Mrk 501, I~Zw~I, and Mrk 335, at $V \leq 14.5$. We find 4 probable ($4.0 \sigma - 4.5 \sigma$) and 4 definite ($5 \sigma - 16 \sigma$) Ly$\alpha$ absorption lines, with equivalent widths $W_{\lambda} = 50 - 200$ m\AA\, corresponding to column densities N(H~I) = $10^{13}$ -- $10^{14}$ cm$^{-2}$, assuming a typical Doppler parameter of $b = 25$ km~s$^{-1}$.
Based on an updated version of the CfA redshift survey (Huchra and Clemens, private communication), most of these Ly$\alpha$ systems appear to be associated with supercluster - sized ``strings'' of galaxies similar to the ``Great Wall''. Toward Mrk 501, the nearest bright galaxy at the redshift of the strongest (200 m\AA) Ly$\alpha$ cloud lies $500 h_{75}^{-1}$~kpc off the line of sight. Models of H~I disks exposed to the intergalactic ionizing radiation field (Dove \& Shull 1994, ApJ, 423 , in press) show that the N(H~I) = $10^{13}$ cm$^{-2}$ contour in a typical spiral galaxy is reached at 100 kpc radial extent. Thus, the Ly$\alpha$ absorbers associated with galaxy-string systems may be the result of H~I in an extended halo, in dwarf satellite galaxies ($M_B > -15$), or in tidally-stripped gas. Most importantly for cosmological origins of baryons, one ($4.3 \sigma$) Ly$\alpha$ absorption line in the spectrum of Mrk 501 lies within the galaxy void in the foreground of the ``Great Wall''. The nearest bright galaxy, to a level $M_B \leq -18.5$ for $H_0 = 75$ km~s$^{-1}$~Mpc$^{-1}$, is more than 5 Mpc away. A pencil-beam survey of faint galaxies to $M_B = -16.0$ finds no galaxy within $100 h_{75}^{-1}$ kpc of the line of sight, at or near the absorber redshift.
