  We present an evaluation of heavy quarkonium states (bottomium and
charmonium) from first principles. We use tree-level QCD (including
relativistic corrections) and the full one-loop potential; nonperturbative
effects are taken into account at the leading order through the contribution of
the gluon condensate <alpha_strong G^2>. We use the values Lambda_QCD= 200
(+80/-60) MeV and <alpha_s G^2> = 0.042 +/- 0.020 GeV^4, but we trade the value
of the quark mass by the masses of J/psi, Upsilon as input. We get good
agreement in what is essentially a zero parameter evaluation for the masses of
the 1S, 2S, 2P states of bottomium, the 1S state for charmonium, and the decay
Upsilon --> e^+ e^-. As outstanding result we obtain the precise determination
of the bottom quark mass (m_b) as well as an estimate of the hyperfine
splitting M(Upsilon)-M(eta_b): m_b(m_b^2) = 4387 (+7/-2) (-3/+4)
(+16/-32)(systematic) MeV, M(Upsilon)-M(eta_b) = 36 (+13/-7)(+3/-6)
(+11/-5)(syst.) MeV, the first error due to that in Lambda_QCD, the second to
that in <alpha_s G^2> (varied independently). For the c quark, we find
m_c(m_c^2)= 1306 (+21/-34)(-6/+6) MeV (up to systematic errors).
