We implement the asymmetric dark matter framework in a supersymmetric model that respects an approximate $U(1)_R$ symmetry, which is broken in such a way that at high temperature the $R$ breaking sector mediate processes in equilibrium, but at the SUSY mass scale, the sparticles asymmetry is frozen. In this framework, the gravitino serves as the dark matter candidate, and its mass is predicted to be $\sim10$ GeV to match the observed relic abundance. We identify several realistic spectra; however, the requirement for the Next-to-Lightest Supersymmetric Particle (NLSP) to decay into the gravitino before Big Bang Nucleosynthesis constrains the viable spectrum to masses above 2 TeV.