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The high-frequency peaked BL Lac PKS 2155-304 at redshift z = 0.116 is a well-known VHE (>100 GeV) gamma -ray emitter. Since 2002 its VHE flux has been monitored using the H.E.S.S. stereoscopic array of imaging atmospheric Cerenkov telescopes in Namibia. During the 2006 July dark period, the average VHE flux was measured to be more than 10 times typical values observed from the object. This article focuses solely on an extreme gamma -ray outburst detected in the early hours of 2006 July 28 (MJD 53,944). The average flux observed during this outburst is I(>200 GeV) = (1.72 plus or minus 0.05 [unk] plus or minus 0.34 [unk]) x 10 super(-9) cm-z s-i, corresponding to similar to 7 times the flux, I(>200 GeV), observed from the Crab Nebula. Peak fluxes are measured with 1 minute timescale resolution at more than twice this average value. Variability is seen up to similar to 600 s in the Fourier power spectrum, and well-resolved bursts varying on timescales of similar to 200 s are observed. There are no strong indications for spectral variability within the data. Assuming the emission region has a size comparable to the Schwarzschild radius of a similar to 10 super(9) M [unk] black hole, Doppler factors greater than 100 are required to accommodate the observed variability timescales.