We report the synthesis and properties of a novel class of nanocrystals with mixed dimensionality: a dot-in-plate core/shell nanostructure. This system was synthesized by growing a flat, disk-shaped, CdS shell on spherical CdSe cores. The anisotropic pressure induced by the shell drastically splits the first exciton fine structure in two: the "heavy hole" and "light hole" states become separated by up to 65 meV. As a result, these nanocrystals exhibit an emission strongly polarized in two dimensions, in the plane perpendicular to the wurtzite crystal c axis. We use polarization measurements on single nanocrystals and ensemble anisotropy studies to confirm the nature and position of the excitonic energy levels. These nanocrystals orient spontaneously when evaporated on a substrate, enabling a precise control of the orientation of their emission dipole.