  During the several months of the outburst of Mkn 501 in 1997 the source has
been monitored in TeV gamma-rays with the HEGRA stereoscopic system of imaging
atmospheric Cherenkov telescopes. Quite remarkably it turned out that the
shapes of the daily gamma-ray energy spectra remained essentially stable
throughout the entire state of high activity despite dramatic flux variations
during this period. The derivation of a long term time-averaged energy
spectrum, based on more than 38,000 detected TeV photons, is therefore
physically meaningful. The unprecedented gamma-ray statistics combined with the
20% energy resolution of the instrument resulted in the first detection of
gamma-rays from an extragalactic source well beyond 10 TeV, and the first high
accuracy measurement of an exponential cutoff in the energy region above 5 TeV
deeply into the exponential regime. From 500 GeV to 24 TeV the differential
photon spectrum is well approximated by a power-law with an exponential cutoff:
dN/dE = N0 (E/1 TeV)^{-alpha} exp(-E/E0), with N0=(10.8 +-0.2(stat) +-2.1(sys))
1E-11/cm^2 s TeV, alpha=1.92 +-0.03(stat) +-0.20(sys), and E0=(6.2 +-0.4(stat)
(-1.5 +2.9)(sys) TeV. We summarize the methods for the evaluation of the energy
spectrum in a broad dynamical range which covers almost two energy decades, and
study in detail the principal sources of systematic errors. We also discuss
several important astrophysical implications of the observed result concerning
the production and absorption mechanisms of gamma-rays in the emitting jet and
the modifications of the initial spectrum of TeV radiation due to its
interaction with the diffuse extragalactic background radiation.
