  The perturbative QCD fragmentation functions for a heavy quark to fragment
into heavy-light mesons are studied in the heavy-quark limit. The fragmentation
functions for S-wave pseudoscalar and vector mesons are calculated to
next-to-leading order in the heavy-quark mass expansion using the methods of
heavy-quark effective theory. The results agree with the $m_b\to \infty$ limit
of the perturbative QCD fragmentation functions for $\bar b$ into $B_c$ and
$B_c^*$. We discuss the application of the perturbative QCD fragmentation
functions as a model for the fragmentation of heavy quarks into heavy-light
mesons. Using this model, we predict the fraction $P_{V}$ of heavy-light mesons
that are produced in the vector meson state as functions of the longitudinal
momentum fraction $z$ and the transverse momentum relative to the jet axis. The
fraction $P_V$ is predicted to vary from about 1/2 at small $z$ to almost 3/4
near $z=1$.
