Feebly interacting thermal relics are promising dark matter candidates. Among them, scenarios of inelastic Dark Matter evade direct detection by suppressed elastic scattering off atomic nuclei. We introduce inelastic Dirac Dark Matter, a new model with two Dirac fermions in the MeV-GeV mass range. At feeble couplings, dark matter can depart from chemical as well as kinetic equilibrium with the Standard Model before freeze-out. In this case, the freeze-out is driven by conversion processes like coscattering, rather than coannihilation. We show that inelastic Dirac relics are consistent with cosmological observations, in particular with nucleosynthesis and the cosmic microwave background. Searches for dark sectors at colliders and fixed-target experiments, in turn, are very sensitive probes. Compared to the strongly constrained pseudo-Dirac scenario, inelastic Dirac Dark Matter offers a new search target for existing and upcoming experiments like Belle II, ICARUS, LDMX and SeaQuest.