  Let $\Sigma$ be a smooth projective complex curve and $\mathfrak{g}$ a simple
Lie algebra of type ${\sf ADE}$ with associated adjoint group $G$. For a fixed
pair $(\Sigma, \mathfrak{g})$, we construct a family of quasi-projective
Calabi-Yau threefolds parameterized by the base of the Hitchin integrable
system associated to $(\Sigma,\mathfrak{g})$. Our main result establishes an
isomorphism between the Calabi-Yau integrable system, whose fibers are the
intermediate Jacobians of this family of Calabi-Yau threefolds, and the Hitchin
system for $G$, whose fibers are Prym varieties of the corresponding spectral
covers. This construction provides a geometric framework for Dijkgraaf-Vafa
transitions of type ${\sf ADE}$. In particular, it predicts an interesting
connection between adjoint ${\sf ADE}$ Hitchin systems and quantization of
holomorphic branes on Calabi-Yau manifolds.
