The extreme synchrotron BL Lac object H 2356-309, located at a redshift of z = 0.165, was observed from June to December 2004 with a total exposure of $\approx$40 h live-time with the HESS (High Energy Stereoscopic System) array of atmospheric-Cherenkov telescopes (ACTs). Analysis of this data set yields, for the first time, a strong excess of 453 $\gamma$-rays (10 standard deviations above background) from H 2356-309, corresponding to an observed integral flux above 200 GeV of I(>200 GeV) = (4.1 $\pm$ 0.5) $\times$ 10-12 cm-2 s-1 (statistical error only). The differential energy spectrum of the source between 200 GeV and 1.3 TeV is well-described by a power law with a normalisation (at 1 TeV) of N0 = (4.1 $\pm$ 0.5) $\times$ 10-13 cm-2 s-1...