  Using an impact parameter tag to select an enriched sample of $Z \rightarrow
b \overline{b}$ events, we have measured the difference between the average
charged multiplicity of $Z \rightarrow b \overline{b}$ and $Z^0 \rightarrow
hadrons$ to be $\overline{n}_{b} - \overline{n}_{had} = 2.24\pm
0.30(\rm{stat.}) \pm 0.33(\rm{syst.})$ tracks per event. {}From this, we have
derived $\overline{n}_{b} - \overline{n}_{uds} = 3.31 \pm 0.41 \pm 0.79.$
Comparing this measurement with those at lower center-of-mass energies, we find
no evidence that $\overline{n}_{b} - \overline{n}_{uds}$ depends on energy.
This result is in agreement with a precise prediction of perturbative QCD, and
supports the notion that QCD remains asymptotically free down to the scale
$M_b^2$.
