  We non-perturbatively calculate the scale dependence of the static axial
current in the Schroedinger functional scheme by means of a recursive
finite-size scaling technique, taking the continuum limit in each step. The
bare current in the O(a) improved theory as well as in the original Wilson
regularization is thus connected to the renormalization group invariant one.
The latter may then be related to the current at the B-scale defined such that
its matrix elements differ from the physical (QCD) ones by O(1/M). At present,
a (probably small) perturbative uncertainty enters in this step. As an
application, we renormalize existing unimproved data on F_B^{bare} and
extrapolate to the continuum limit. We also study an interesting function
h(d/L,u) derived from the Schroedinger functional amplitude describing the
propagation of a static quark-antiquark pair.
