  For a complex simple simply connected Lie group $G$, and a compact Riemann
surface $C$, we consider two sorts of families of flat $G$-connections over
$C$. Each family is determined by a point ${\mathbf u}$ of the base of
Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf
u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The
other family $\nabla_{R,\zeta,{\mathbf u}}$ is built from solutions of
Hitchin's equations, and depends on $\zeta \in {\mathbb C}^\times, R \in
{\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $\zeta = \hbar R$,
we have $\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This
establishes and generalizes a conjecture formulated by Gaiotto.
