Observations of the shell-type supernova remnant SN1006 have been carried out with the HESS system of Cherenkov telescopes during 2003 (18.2 h with two operating telescopes) and 2004 (6.3 h with all four telescopes). No evidence for TeV $γ$-ray emission from any compact or extended region associated with the remnant is seen and resulting upper limits at the 99.9% confidence level are up to a factor 10 lower than previously-published fluxes from CANGAROO. For SN1006 at its current epoch of evolution we define limits for a number of important global parameters. Upper limits on the $γ$-ray luminosity (for E = 0.26 to 10 TeV, distance d = 2 kpc) of L$γ$ 25 $μ$G results when considering the synchrotron/inverse-Compton framework for the observed X-ray flux and $γ$-ray upper limits.