  A formula for an improved spin--dependent potential between a heavy quark and
a heavy antiquark was developed using Heavy Quark Effective Theory techniques.
The leading logarithmic quark mass terms emerging from loop contributions were
explicitly extracted and summed up. There is no renormalization scale
dependence in this new formula and it includes both the Eichten--Feinberg
formula as well as the one--loop QCD result as special cases. The hyperfine
splittings were calculated using the spin--spin part of this improved formula.
For charmonium the $J/\psi$-$\eta_c$ $S$-wave splitting agrees well with the
data while the energy difference $\Delta M_P$ between the center of gravity of
the $1^3P_{0,1,2}$ states and the $1^1P_1$ state has the correct sign but is
somewhat larger than the experimental value; however, there are also several
other contributions to $\Delta M_P$ of comparable magnitude ($\sim 1$ MeV),
which we discuss. The corresponding mass differences for the $b\bar{b}$ and
$b\bar{c}$ mesons are also predicted.
