  We extract (for the first time) the correlated values of the running masses
m_c and m_b from M_Bc using QCD Laplace sum rules (LSR) within stability
criteria where pertubative (PT) expressions at N2LO and non-perturbative (NP)
gluon condensates at LO are included. Allowing the values of m_{c,b}(m_{c,b})
to move inside the enlarged range of recent estimates from charmonium and
bottomium sum rules (Table 1) obtained using similar stability criteria, wee
deduce : m_c(m_c) = 1286(16) MeV and m_b(m_b) = 4208(8) MeV. Combined with
previous estimates (Table 2), we deduce a tentative QCD Spectral Sum Rules
(QSSR) average m_c(m_c) = 1266(6) MeV and m_b(m_b) = 4197(8) MeV, where the
errors come from the precise determinations from J/psi and Upsilon sum rules.
As a result, we present an improved prediction of f_B_c =371(17)MeV and the
tentative upper bound f_B_c(2S)<139(6) MeV, which are useful for a further
analysis of B_c-decays.
