  We calculate the cross section for $e^+ e^-$ annihilation into three jets for
massive quarks at next-to-leading order in perturbative QCD, both on and off
the $Z$ resonance. Our computation allows the implementation of any jet
clustering algorithm. We give results for the three-jet cross section involving
$b$ quarks for the JADE and Durham algorithm at c.m. energies $\sqrt s = m_Z$.
We also discuss a three-jet observable that is sensitive to the mass of the $b$
quark.
