Aims. Recent theoretical predictions of the lowest very high energy (VHE) luminosity of SN 1006 are only a factor 5 below the previously published HESS upper limit, thus motivating further in-depth observations of this source. Methods. Deep observations at VHE energies (above 100 GeV) were carried out with the high energy stereoscopic system (HESS) of Cherenkov Telescopes from 2003 to 2008. More than 100 h of data have been collected and subjected to an improved analysis procedure. Results. Observations resulted in the detection of VHE γ-rays from SN 1006. The measured γ-ray spectrum is compatible with a power-law, the flux is of the order of 1% of that detected from the Crab Nebula, and is thus consistent with the previously established HESS upper limit. The source exhibits a bipolar morphology, which is strongly correlated with non-thermal X-rays. Conclusions. Because the thickness of the VHE-shell is compatible with emission from a thin rim, particle acceleration in shock waves is likely to be the origin of the γ-ray signal. The measured flux level can be accounted for by inverse Compton emission, but a mixed scenario that includes leptonic and hadronic components and takes into account the ambient matter density inferred from observations also leads to a satisfactory description of the multi-wavelength spectrum.