Silicon carbide (SiC) is an excellent substrate for growth and manipulation of large scale, high quality epitaxial graphene. On the carbon face (the ( 111 ) or (000 1) face, depending on the polytype), the onset of graphene growth is intertwined with the formation of several competing surface phases, among them a (3×3) 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 (3×3)-SiC-( 111 ) reconstruction, derived from an 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 (3×3) unit cell formally unsaturated. Simulated scanning tunneling microscopy (STM) images are in excellent agreement with previously reported experimental STM images.