We present results of a 40 ks ROSAT pointed observation of the Hyades cluster. The limiting L(sub x) is approximately = 2 x 10(exp 27) ergs/sec at field center, increasing to approximately = 2 x 10(exp 28) ergs/sec at 40 min off-axis. This represents the most sensitive X-ray observation to date in the Hyades region. More than 30 sources have been detected in the Position Sensitive Proportional Counter (PSPC) standard processing, of which 15 are Hyades members, five are cluster candidates that are likely non-members, four are foreground or background stars, and the remainder are unidentified. One Hyad, VB 173 (= VA 276), lies in a confused region, but is detected as a distinct source in the soft band only (E approximately less than 0.5 keV). We report upper limits for four other Hyades members in our field, all M dwarfs. Of the 16 Hyades detections, two represent the optically faintest members seen to date in X-rays; VA 260 (V = 16.68) and VA 368 (V = 16.25). These are both M dwarfs of mass approximately = 0.15-0.2 solar mass and are fully convective stars according to current theory. Analysis of X-ray light curves using 1 ks bins indicates some variability in the strongest sources and a possible flare in VA 383. Two Hyades stars, VB 141 and VB 71, were also detected with the co-aligned Wide Field Camera (WFC) EUV instrument. VB 141, the second brightest X-ray source in the Hyades, remains an enigma: a rapidly rotating FO star with a fainter, long-period companion, this object has an X-ray spectrum indicative of strong coronal activity. X-ray pulse-height analysis demonstrates that coronal models with at least two temperatures are required for most of the stronger X-ray sources. The ROSAT X-ray spectra generally require higher temperatures for the hotter component in the M dwarfs compared to the F-G dwarfs.