  For quasianalytic Denjoy--Carleman differentiable function classes $C^Q$
where the weight sequence $Q=(Q_k)$ is log-convex, stable under derivations, of
moderate growth and also an $\mathcal L$-intersection (see 1.6), we prove the
following: The category of $C^Q$-mappings is cartesian closed in the sense that
$C^Q(E,C^Q(F,G))\cong C^Q(E\times F, G)$ for convenient vector spaces.
Applications to manifolds of mappings are given: The group of
$C^Q$-diffeomorphisms is a regular $C^Q$-Lie group but not better.
