  We present a detailed study of the ionization fraction of the Barnard 68
pre-stellar core, using millimeter H13CO+ and DCO+ lines observations. These
observations are compared to the predictions of a radiative transfer model
coupled to a chemical network that includes depletion on grains and gas phase
deuterium fractionation. Together with previous observations and modelling of
CO and isotopologues, our H13CO+ and DCO+ observations and modelling allow to
place constraints on the metal abundance and the cosmic ionization rate. The
H13CO+ emission is well reproduced for metals abundances lower than 3e-9 and a
standard cosmic ray ionization rate. However, the observations are also
consistent with a complete depletion of metals, i.e. with cosmic rays as the
only source of ionization at visual extinctions greater than a few Av . The
DCO+ emission is found to be dependent of the ortho to para H2 ratio, and
indicates a ratio of ~1e-2. The derived ionization fraction is about 5e-9 with
respect to H nuclei, which is about an order of magnitude lower than the one
observed in the L1544 core. The corresponding ambipolar diffusion timescale is
found to be an order of magnitude larger than the free fall timescale at the
center of the core. The inferred metal abundance suggests that magnetically
inactive regions (dead zones) are present in protostellar disks.
