  We consider the free energy $W[J] = W_k(H)$ of QCD coupled to an external
source $J_\mu^b(x) = H_\mu^b \cos(k \cdot x)$, where $H_\mu^b$ is, by analogy
with spin models, an external "magnetic" field with a color index that is
modulated by a plane wave. We report an optimal bound on $W_k(H)$ and an exact
asymptotic expression for $W_k(H)$ at large $H$. They imply confinement of
color in the sense that the free energy per unit volume $W_k(H)/V$ and the
average magnetization $m(k, H) ={1 \over V} {\p W_k(H) \over \p H}$ vanish in
the limit of constant external field $k \to 0$. Recent lattice data indicate a
gluon propagator $D(k)$ which is non-zero, $D(0) \neq 0$, at $k = 0$. This
would imply a non-analyticity in $W_k(H)$ at $k = 0$. We also give some general
properties of the free energy $W(J)$ for arbitrary $J(x)$. Finally we present a
model that is consistent with the new results and exhibits (non)-analytic
behavior. Direct numerical tests of the bounds are proposed.
