  We revisit the sum-rule extraction of the decay constants of the D, Ds, B,
and Bs mesons from the two-point correlator of heavy-light pseudoscalar
currents. We use the operator product expansion for this correlator expressed
in terms of the MSbar heavy-quark mass, for which the perturbative expansion
exhibits a reasonable convergence. Our main emphasis is laid on the control
over the uncertainties in the decay constants, related both to the input QCD
parameters and to the limited accuracy of the method of sum-rules. This becomes
possible due to the application of our procedure of extracting hadron
observables that involves as novel feature dual thresholds depending on the
Borel parameter. For charmed mesons, we find the decay constants f_D=206.2\pm
7.3(OPE)\pm 5.1(syst) MeV and f_Ds=245.3\pm 15.7(OPE)\pm 4.5(syst) MeV. For
beauty mesons, the decay constants turn out to be extremely sensitive to the
precise value of mb(mb). By requiring our sum-rule estimate to match the
average of the lattice results for f_B, a very accurate value mb(mb)=4.245\pm
0.025 GeV is extracted, leading to f_B=193.4\pm 12.3(OPE)\pm 4.3(syst) MeV and
f_Bs=232.5\pm 18.6(OPE)\pm 2.4(syst) MeV.
