We report infrared spectroscopic observations of HD 105, a nearby ($sim 40$ pc) and relatively young ($sim 30$ Myr) G0 star with excess infrared continuum emission, which has been modeled as arising from an optically thin circumstellar dust disk with an inner hole of size $gtrsim 13$ AU. We have used the high spectral resolution mode of the Infrared Spectrometer (IRS) on the Spitzer Space Telescope to search for gas emission lines from the disk. The observations reported here provide upper limits to the fluxes of H$_2$ S(0) 28$mu$m, H$_2$ S(1) 17$mu$m, H$_2$ S(2) 12 $mu$m, [FeII] 26$mu$m, [SiII] 35$mu$m, and [SI] 25$mu$m infrared emission lines. The H$_2$ line upper limits directly place constraints on the mass of warm molecular gas in the disk: $M({rm H_2})< 4.6$, 3.8$times 10^{-2}$, and $3.0times 10^{-3}$ M$_J$ at $T= 50$, 100, and 200 K, respectively. We also compare the line flux upper limits to predictions from detailed thermal/chemical models of various gas distributions in the disk. These comparisons indicate that if the gas distribution has an inner hole with radius $r_{i,gas}$, the surface density at that inner radius is limited to values ranging from $lesssim 3$ gm cm$^{-2}$ at $r_{i,gas}=0.5$ AU to 0.1 gm cm$^{-2}$ at $r_{i,gas}= 5-20$ AU. These values are considerably below the value for a minimum mass solar nebula, and suggest that less than 1 M$_J$ of gas (at any temperature) exists in the 1-40 AU planet-forming region. Therefore, it is unlikely that there is sufficient gas for gas giant planet formation to occur in HD 105 at this time.
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