The Vela supernova remnant (SNR) is a complex region containing a number of sources of non-thermal radiation. The inner section of this SNR, within 2 degrees of the pulsar PSR B0833−45, has been observed by the HESS γ-ray atmospheric Cherenkov detector in 2004 and 2005. A strong signal is seen from an extended region to the south of the pulsar, within an integration region of radius 0.8• around the position (α = 08 h 35 m 00 s , δ = −45• 36 J2000.0). The excess coincides with a region of hard X-ray emission seen by the ROSAT and ASCA satellites. The observed energy spectrum of the source between 550 GeV and 65 TeV is well fit by a power law function with photon index Γ = 1.45 ± 0.09 stat ± 0.2 sys and an exponential cutoff at an energy of 13.8 ± 2.3 stat ± 4.1 sys TeV. The integral flux above 1 TeV is (1.28 ± 0.17 stat ± 0.38 sys ) × 10 −11 cm −2 s −1 . This result is the first clear measurement of a peak in the spectral energy distribution from a VHE γ-ray source, likely related to inverse Compton emission. A fit of an Inverse Compton model to the HESS spectral energy distribution gives a total energy in non-thermal electrons of ∼2 × 10 45 erg between 5 TeV and 100 TeV, assuming a distance of 290 parsec to the pulsar. The best fit electron power law index is 2.0, with a spectral break at 67 TeV.