  We calculate the process-independent fragmentation functions for a $\bar b$
antiquark to fragment into longitudinally and transversely polarized $B_c^*$
($^3S_1$) mesons to leading order in the QCD strong coupling constant. In the
special case of equal quark mass we recover previous results for the
fragmentation of $c\to \psi$ and $b\to\Upsilon$. Various spin asymmetry
parameters are defined as measures of the relative population of the
longitudinally and transversely polarized vector meson states. In the heavy
quark mass limit $m_b \to \infty$ our polarized fragmentation functions obey
heavy quark spin symmetry, we therefore apply them as a model to describe the
fragmentation of charm and bottom into heavy-light mesons like $D^*$ and $B^*$.
The spin asymmetry parameter, $\alpha(z)$, is consistent with the existing CLEO
data for $D^*$. The scaling behavior of $\langle z \rangle$ is studied in
detail. We find excellent agreement between the predictions of $\langle z
\rangle$ from our fragmentation functions and the experimental data for $D^*$
and $B^*$ from the LEP, CLEO, and ARGUS detectors. Finally, we also point out
that the spin asymmetry depends significantly on the transverse momentum
$p_\perp$ of the vector mesons relative to the fragmentation axis.
