Swift XRT Observations of the Possible Dark Galaxy VIRGOHI 21

\swift \xrt observations of the \hi line source \VV were performed on April 22 and April 26, 2008 for a total exposure time of 9.2ks. This is the first pointed X-ray observation of \VV, a putative dark galaxy in the Virgo cluster, and no photons were detected from this source. The non-detection of extended X-ray emission within the angular extent of the \hi source corresponds to a 99% confidence upper limit of $2.1 \times 10^{-14}$\ergcms in the 0.3--2.0 keV band. The equivalent upper limit to the amount of diffuse hot gas associated with \VV is in the range $4\times 10^{7} - 2 \times 10^8$ \msun for a hot gas temperature between 0.1 and 1 keV. The non-detection also corresponds to a 99%-confidence upper limit on the flux from a point-like source of $8\times 10^{-15}$\ergcms in the 0.3--2.0 keV band. We discuss the constraints on the nature of \VV imposed by these observations and the theoretical implications of these results.