From about 4 million hadronic $\rm Z^0$ decays recorded by the OPAL
detector on and near to the $\rm Z^0$ resonance, we select a sample of more
than $570\, 000$ inclusively reconstructed B mesons.
Orbitally-excited mesons ${\rm B}^*_J$ are reconstructed using $\rm B\pi^\pm$
combinations. Independently, $\rm B^*$ mesons are reconstructed using
the decay $\rm B^*\rightarrow B\gamma$. The selected $\rm B^*$ candidates 
are used to
obtain samples enriched or depleted in the decay 
${\rm B}^*_J\rightarrow\rm B^*\pi^\pm (X)$, where
(X) refers to decay modes with or without additional
accompanying decay particles. From the number of signal candidates in
the $\rm B\pi^\pm$ mass spectra of these two samples, we perform the first
measurement of the branching ratio of orbitally-excited B mesons
decaying into $\rm B^*\pi(X)$:
\[
{\rm BR(B}^*_J\rightarrow\rm B^*\pi(X)) = 0.85^{\,+0.26}_{\,-0.27} \pm 0.12 \, ,
\]
where the first error is statistical and the second systematic.