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Accurate determinations of the MS-bar b-quark mass $m_b(m_b)$ from$\sigma(e^+e^-\to{\rm hadrons})$ experimental data currently contain threecomparable sources of uncertainty; the experimental uncertainty from moments ofthis cross-section, the uncertainty associated with $\alpha_s(M_z)$, and thetheoretical uncertainty associated with the renormalization scale. Throughresummation of all logarithmic terms explicitly determined in the perturbativeseries by the renormalization-group (RG) equation, it is shown that therenormalization-scale dependence is virtually eliminated as a source oftheoretical uncertainty in $m_b(m_b)$. This resummation also reduces theestimated effect of higher-loop perturbative contributions, further reducingthe theoretical uncertainties in $m_b(m_b)$. Furthermore, such resummationtechniques improve the agreement between the values of the MS-bar b-quark massextracted from the various moments of $R(s)=\sigma(e^+e^-\to{\rmhadrons})/\sigma_{pt}$ [$\sigma_{pt}=4\pi\alpha^2/(3s)$], obviating the need tochoose an optimummoment for determining $m_b(m_b)$. Resummation techniques arealso shown to reduce renormalization-scale dependence in the relation betweenb-quark MS-bar and pole mass and in the relation between the pole and $1S$mass.Comment: 19 pages, latex2e, 6 eps figures contained in latex file. Errors  corrected in equations (20)--(22

IMPROVEMENT OF RENORMALIZATION-SCALE UNCERTAINTIES WITHIN EMPIRICAL DETERMINATIONS OF THE b-QUARK MASS