  The BL Lac object Mrk 501 went into a very high state of activity during
1997, both in VHE gamma-rays and X-rays. We present here results from
observations at energies above 250 GeV carried out between March and October
1997 with the CAT Cerenkov imaging Telescope. The average differential spectrum
between 30 GeV and 13 TeV shows significant curvature and is well represented
by phi_0 * E_TeV^{-(alpha + beta*log10(E_TeV))}, with: phi_0 = 5.19 +/- 0.13
{stat} +/- 0.12 {sys-MC} +1.66/-1.04 {sys-atm} * 10^-11 /cm^2/s/TeV alpha =
2.24 +/- 0.04 {stat} +/- 0.05 {sys} beta = 0.50 +/- 0.07 {stat} (negligible
systematics). The TeV spectral energy distribution of Mrk 501 clearly peaks in
the range 500 GeV-1 TeV. Investigation of spectral variations shows a
significant hardness-intensity correlation with no measurable effect on the
curvature. This can be described as an increase of the peak TeV emission energy
with intensity. Simultaneous and quasi-simultaneous CAT VHE gamma-ray and
BeppoSAX hard X-ray detections for the highest recorded flare on 16th April and
for lower-activity states of the same period show correlated variability with a
higher luminosity in X-rays than in gamma-rays. The observed spectral energy
distribution and the correlated variability between X-rays and gamma-rays, both
in amplitude and in hardening of spectra, favour a two-component emission
scheme where the low and high energy components are attributed to synchrotron
and inverse Compton (IC) radiation, respectively.
