In the Minimal Supersymmetric Standard Model (MSSM), the scalar neutrino $tilde{ u}_L$ has odd R parity, yet it has long been eliminated as a dark-matter candidate because it scatters elastically off nuclei through the $Z$ boson, yielding a cross section many orders of magnitude above the experimental limit. We show how it can be reinstated as a dark-matter candidate by splitting the masses of its real and imaginary parts in an extension of the MSSM with scalar triplets. As a result, radiative Majorana neutrino masses are also generated. In addition, decays of the scalar triplets relate the abundance of this asymmetric dark matter to the baryon asymmetry of the Universe through leptogenesis.