  Mkn 421 was observed during a high flux state for nine nights in April and
May 2004 with the fully operational High Energy Stereoscopic System (H.E.S.S.)
in Namibia. The observations were carried out at zenith angles of
60$^\circ$--65$^\circ$, which result in an average energy threshold of 1.5 TeV
and a collection area reaching 2~km$^2$ at 10~TeV. Roughly 7000 photons from
Mkn~421 were accumulated with an average gamma-ray rate of 8 photons/min. The
overall significance of the detection exceeds 100 standard deviations. The
light-curve of integrated fluxes above 2~TeV shows changes of the diurnal flux
up to a factor of 4.3. For nights of high flux, intra-night variability is
detected with a decay time of less than 1 hour. The time averaged energy
spectrum is curved and is well described by a power-law with a photon index
$\egamm$ and an exponential cutoff at $\ecut$~TeV and an average integral flux
above 2~TeV of 3 Crab flux units. Significant variations of the spectral shape
are detected with a spectral hardening as the flux increases. Contemporaneous
multi-wavelength observations at lower energies (X-rays and gamma-rays above
$\approx 300$~GeV) indicate smaller relative variability amplitudes than seen
above 2~TeV during high flux state observed in April 2004.
