The atomic and electronic structures of a graphene layer on top of the $(2\ifmmode\times\else\texttimes\fi{}2)$ reconstruction of the SiC $(000\overline{1})$ surface are studied from ab initio calculations. At variance with the (0001) face, no C buffer layer is found here. Si adatoms passivate the substrate surface so that the very first C layer presents a linear dispersion characteristic of graphene. A small graphene-substrate interaction remains in agreement with scanning tunneling experiments [F. Hiebel, P. Mallet, F. Varchon, L. Magaud, and J.-Y. Veuillen, Phys. Rev. B 78, 153412 (2008)]. The stacking geometry has little influence on the interaction which explains the rotational disorder observed on this face.