  Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its
Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be
the dual Lie bialgebra. By dualizing the quantum double construction - via
formal Hopf algebras - we construct a new quantum group
$U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its
restricted and unrestricted integer forms and their specializations at roots of
1 (in particular, their classical limits), we prove that
$U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and
$\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we
construct new quantum Frobenius morphisms. The whole picture extends to the
untwisted affine case the results known for quantum groups of finite type.
