We present results from a 36-ksec observation of the core of the Pleiades open cluster using ACIS-I on the Chandra X-ray Observatory. We have detected 57 sources, most of which do not have previously known optical counterparts. Follow-up photometry indicates that many of the detections are likely to be AGNs, in accordance with extragalactic source counts, but some of the sources may be previously undiscovered low-mass members of the Pleiades. We discuss our dataset and our findings about X-ray emission from early-type stars as well as very late type stars. In particular, the large X-ray fluxes, lack of variability, and hardness ratios of the four Pleiades B6 IV -- F4 V stars suggest a tentative conclusion that Pleiades stars in this spectral type range are intrinsic X-ray sources rather than previously unknown binaries in which the X-ray emission is from a late-type companion. Also the sensitivity of Chandra allowed us to detect nonflare X-ray emission from late-M stars.