Context : The Crab nebula was observed with the H .E.S.S . stereoscopic Cherenkov-telescope array between October 2003 and January 2005 for a total of 22.9 hours ( after data quality selection ) .
This period of time partly overlapped with the commissioning phase of the experiment ; observations were made with three operational telescopes in late 2003 and with the complete 4 telescope array in January – February 2004 and October 2004 – January 2005 .

Aims : Observations of the Crab nebula are discussed and used as an example to detail the flux and spectral analysis procedures of H .E.S.S . .
The results are used to evaluate the systematic uncertainties in H .E.S.S . flux measurements .

Methods : The Crab nebula data are analysed using standard H .E.S.S . analysis procedures , which are described in detail .
The flux and spectrum of \gamma -rays from the source are calculated on run-by-run and monthly time-scales , and a correction is applied for long-term variations in the detector sensitivity .
Comparisons of the measured flux and spectrum over the observation period , along with the results from a number of different analysis procedures are used to estimate systematic uncertainties in the measurements .

Results : The data , taken at a range of zenith angles between 45 ^ { \circ } and 65 ^ { \circ } , show a clear signal with over 7500 excess events .
The energy spectrum is found to follow a power law with an exponential cutoff , with photon index \Gamma = 2.39 \pm 0.03 _ { \textrm { \tiny { stat } } } and cutoff energy E _ { c } = ( 14.3 \pm 2.1 _ { \textrm { \tiny { stat } } } ) \textrm { TeV } between 440 GeV and 40 TeV .
The observed integral flux above 1 TeV is ( 2.26 \pm 0.08 _ { \textrm { \tiny { stat } } } ) \times 10 ^ { -11 } \textrm { cm } ^ { -2 } \textrm { s } % ^ { -1 } .
The estimated systematic error on the flux measurement is estimated to be 20 % , while the estimated systematic error on the spectral slope is 0.1 .

Conclusions :