  We present the results of a perturbative calculation to match the axial and
vector currents for semileptonic $B\to D^{(\ast)}$ decays in lattice NRQCD to
the continuum \MSb scheme. The matching is performed to
$O(\alpha_s\frac{\vec{p}}{M})$ in Feynman gauge and in the on-shell
renormalisation scheme.
  The spatial and temporal components of the currents renormalise differently;
to this order the matching involves a straightforward renormalisation for the
$V_0$ and $A_k$ currents, and a rank two and four mixing matrix for the $A_0$
and $V_k$ currents respectively. The resultant one loop corrections are of
$O(5%)$, boding well for the accuracy of forthcoming simulations.
