  In this talk we discuss results of a new extraction of the MS-bar charm quark
mass using relativistic QCD sum rules at O(as**3) based on moments of the
vector and the pseudoscalar current correlators and using the available
experimental measurements from e+e- collisions and lattice results,
respectively. The analysis of the perturbative uncertainties is based on
different implementations of the perturbative series and on independent
variations of the renormalization scales for the mass and the strong coupling
following a work we carried out earlier. Accounting for the perturbative series
that result from this double scale variation is crucial since some of the
series can exhibit extraordinarily small scale dependence, if the two scales
are set equal. The new aspect of the work reported here adresses the problem
that double scale variation might also lead to an overestimate of the
perturbative uncertainties. We supplement the analysis by a convergence test
that allows to quantify the overall convergence of QCD perturbation theory for
each moment and to discard series that are artificially spoiled by specific
choices of the renormalization scales. We also apply the new method to an
extraction of the MS-bar bottom quark mass using experimental moments that
account for a modeling uncertainty associated to the continuum region where no
experimental data is available. We obtain m_c(m_c) = 1.287 +- 0.020 GeV and
m_b(m_b) = 4.167 +- 0.023 GeV.
