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 (HESS) in Namibia. The observations were carried out at zenith angles of 60°-65°, which result in an average energy threshold of 1.5 TeV and a collection area reaching 2 km2 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 h. The time averaged energy spectrum is curved and is well described by a power-law with a photon index $Γ$=2.1$±$0.1stat$±$0.3sys and an exponential cutoff at Ec=3.1(+0.5 -0.4)stat$±$0.9sys 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 $≈$300 GeV) indicate smaller relative variability amplitudes than seen above 2 TeV during high flux state observed in April 2004.