The BL Lacertae object Markarian 501, one of only three extragalactic sources (with Mrk 421 and 1ES 2344+514) so far detected at TeV energies, was observed with the BeppoSAX satellite in 1997 April 7, 11, and 16 during a phase of high activity at TeV energies, as monitored with the Whipple, HEGRA, and CAT Cherenkov telescopes. Over the whole 0.1-200 keV range, the spectrum was exceptionally hard (α≤1, with Fν ∝ ν−α), indicating that the X-ray power output peaked at (or above) ~100 keV. This represents a shift of at least 2 orders of magnitude with respect to previous observations of Mrk 501, a behavior never seen before in this or any other blazar. The overall X-ray spectrum hardens with increasing intensity, and at each epoch it is softer at larger energies. The correlated variability from soft X-rays to the TeV band points to models in which the same population of relativistic electrons produces the X-ray continuum via synchrotron radiation and the TeV emission by inverse Compton scattering of the synchrotron photons or other seed photons. For the first time in any blazar, the synchrotron power is observed to peak at hard X-ray energies. The large shift of the synchrotron peak frequency with respect to previous observations of Mrk 501 implies that intrinsic changes in the relativistic electron spectrum caused the increase in emitted power. Due to the very high electron energies, the inverse Compton process is limited by the Klein-Nishina regime. This implies a quasi-linear (as opposed to quadratic) relation of the variability amplitude in the TeV and hard X-ray ranges (for the synchrotron self-Compton model) and an increase of the inverse Compton peak frequency smaller than that of the synchrotron peak frequency.