The imaging atmospheric Cherenkov array H.E.S.S. recently discovered an extended source in the 0.4$-$10 TeV energy range, HESS J1303-631. We obtained a 5 ks observation with the ACIS-I array on the Chandra X-ray observatory that does not reveal an obvious compact or diffuse X-ray counterpart. Archival ROSAT images are also blank in this region. Although there are several radio pulsars within the field of HESS J1303-631, none is detected in X-rays to a flux limit of $<5 times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$, and none is a likely counterpart on energetic grounds. Over the entire $17^{prime} times 17^{prime}$ ACIS-I field, we place an upper limit of $<5.4 times 10^{-12}$ ergs cm$^{-2}$ s$^{-1}$ on the excess diffuse flux in the 2-10 keV band. One hard point-source with flux $approx 4 times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$ lies within $0.^{prime}5$ of the centroid of the TeV emission. These exploratory observations suggest that deeper pointings with Chandra and XMM are needed before we can learn more about the nature of HESS J1303-631. Its similarity to the unidentified source TeV J2032+4130 indicates the probable existence of a new class of high-energy source in the Galactic plane that originates from young, massive stars or their supernova remnants.
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