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In this talk we discuss results of a new extraction of the MS-bar charm quarkmass using relativistic QCD sum rules at O(as**3) based on moments of thevector and the pseudoscalar current correlators and using the availableexperimental measurements from e+e- collisions and lattice results,respectively. The analysis of the perturbative uncertainties is based ondifferent implementations of the perturbative series and on independentvariations of the renormalization scales for the mass and the strong couplingfollowing a work we carried out earlier. Accounting for the perturbative seriesthat result from this double scale variation is crucial since some of theseries can exhibit extraordinarily small scale dependence, if the two scalesare set equal. The new aspect of the work reported here adresses the problemthat double scale variation might also lead to an overestimate of theperturbative uncertainties. We supplement the analysis by a convergence testthat allows to quantify the overall convergence of QCD perturbation theory foreach moment and to discard series that are artificially spoiled by specificchoices of the renormalization scales. We also apply the new method to anextraction of the MS-bar bottom quark mass using experimental moments thataccount for a modeling uncertainty associated to the continuum region where noexperimental data is available. We obtain m_c(m_c) = 1.287 +- 0.020 GeV andm_b(m_b) = 4.167 +- 0.023 GeV.

Charm and Bottom Masses from Sum Rules with a Convergence Test