  Silicon carbide (SiC) is an excellent substrate for growth and manipulation
of large scale, high quality epitaxial graphene. On the carbon face (the
($\bar{1}\bar{1}\bar{1}$) or $(000\bar{1}$) face, depending on the polytype),
the onset of graphene growth is intertwined with the formation of several
competing surface phases, among them a (3$\times$3) precursor phase suspected
to hinder the onset of controlled, near-equilibrium growth of graphene. Despite
more than two decades of research, the precise atomic structure of this phase
is still unclear. We present a new model of the
(3$\times$3)-SiC-($\bar{1}\bar{1}\bar{1}$) reconstruction, derived from an {\it
ab initio} random structure search based on density functional theory including
van der Waals effects. The structure consists of a simple pattern of five Si
adatoms in bridging and on-top positions on an underlying, C-terminated
substrate layer, leaving one C atom per (3$\times$3) unit cell formally
unsaturated. Simulated scanning tunneling microscopy (STM) images are in
excellent agreement with previously reported experimental STM images.
