NGC 6251 has been observed in low-energy X-rays with the Position Sensitive Proportional Counter on board ROSAT. No X-ray emission is seen from the 45 long radio jet of this galaxy, suggesting that the electron spectrum is cut off for electron energies greater than E(max) almost-equal-to 10(12) eV, but a bright X-ray source is coincident with the center of the galaxy. Most of the counts in this source are in a component of small angular size, FWHM < 4'', corresponding to a linear size less than 3 kpc, although there is evidence for the presence of a low surface brightness halo with FWHM approximately 3'. If this halo is thermal bremsstrahlung of gas near NGC 6251, this gas cannot confine the arcminute-scale radio jet by simple (static) gas pressure. The spectrum of the X-ray core source in NGC 6251 is a bad match to a single-component thermal model and, when fitted by a simple power-law model, requires the presence of an absorbing column far in excess of the Galactic or intrinsic columns. The nuclear spectrum can be well fitted by a composite model with two (or more) different components: a plausible decomposition consists of a flat-spectrum (energy index alpha(x) almost-equal-to 0.27) power-law component from a mini-active galactic nucleus (AGN) plus an 0.5 keV thermal component. The power-law spectral index is close to the high-frequency radio spectral index of the VLBI core, supporting a synchro-self-Compton model for this component of the X-ray emission. The temperature of the thermal component is characteristic of the temperatures of gas found in other elliptical galaxies, and the ps must be in a cooling flow. The infall of material into the nucleus is sufficiently rapid to power the mini-AGN component, if the mass of the central black hole exceeds 5 x 10(4) M.. Unification schemes that relate BL Lac objects and radio galaxies (e.g., Padovani & Urry 1990) require a component of unbeamed X-ray emission: it is possible that the type of thermal gas that we find in NGC 6251 may produce the necessary unbeamed X-ray com ponent.