  This note presents an analysis of the potential of future high-energy
electron-positron colliders to measure the $b$-quark mass. We perform a
full-simulation study of the measurement of the ratio of the three-jet rates in
events with $b\bar{b}(g)$ and $q\bar{q}(g)$ production, $R_{3}^{bl}$, and
assess the dominant uncertainties, including theory and experimental systematic
uncertainties. We find that the ILC "Higgs factory" stage, with an integrated
luminosity of 2 ab$^{-1}$ at $\sqrt{s}=$ 250 GeV can measure the $b-$quark
$bar{MS}$ mass at a scale of 250 GeV ($m_b(250~$GeV$)$) with a precision of 1
GeV. From this result we extrapolate the potential of the GigaZ run running at
$\sqrt{s}= m_Z$. We expect $m_b(m_Z)$ can be determined with an 0.12 GeV
uncertainty, exceeding the precision of the LEP and SLD measurements by a
factor $\sim$3.
