We have carried out calculations of ionization equilibrium and deuterium fractionation for conditions appropriate to a completely depleted, low mass pre-protostellar core, where heavy elements such as C, N, and O have vanished from the gas phase and are incorporated in ice mantles frozen on dust grain surfaces. We put particular emphasis on the interpretation of recent observations of H₂D⁺ towards the centre of the prestellar core L 1544 (Caselli et al. \cite{Caselli03}) and also compute the ambipolar diffusion timescale. We consider explicitly the ortho and para forms of H_2, H₃⁺, and H₂D⁺. Our results show that the ionization degree under such conditions depends sensitively on the grain size distribution or, more precisely, on the mean grain surface area per hydrogen nucleus. Depending upon this parameter and upon density, the major ion may be H⁺, H₃⁺, or D₃⁺. We show that the abundance of ortho-H₂D⁺ observed towards L 1544 can be explained satisfactorily in terms of a complete depletion model and that this species is, as a consequence, an important tracer of the kinematics of prestellar cores.