  Accurate determinations of the MS-bar b-quark mass $m_b(m_b)$ from
$\sigma(e^+e^-\to{\rm hadrons})$ experimental data currently contain three
comparable sources of uncertainty; the experimental uncertainty from moments of
this cross-section, the uncertainty associated with $\alpha_s(M_z)$, and the
theoretical uncertainty associated with the renormalization scale. Through
resummation of all logarithmic terms explicitly determined in the perturbative
series by the renormalization-group (RG) equation, it is shown that the
renormalization-scale dependence is virtually eliminated as a source of
theoretical uncertainty in $m_b(m_b)$. This resummation also reduces the
estimated effect of higher-loop perturbative contributions, further reducing
the theoretical uncertainties in $m_b(m_b)$. Furthermore, such resummation
techniques improve the agreement between the values of the MS-bar b-quark mass
extracted from the various moments of $R(s)=\sigma(e^+e^-\to{\rm
hadrons})/\sigma_{pt}$ [$\sigma_{pt}=4\pi\alpha^2/(3s)$], obviating the need to
choose an optimummoment for determining $m_b(m_b)$. Resummation techniques are
also shown to reduce renormalization-scale dependence in the relation between
b-quark MS-bar and pole mass and in the relation between the pole and $1S$
mass.
