  We study the Hubbard-Holstein model, which includes both the
electron-electron and electron-phonon interactions characterized by $U$ and
$g$, respectively, employing the dynamical mean-field theory combined with
Wilson's numerical renormalization group technique. A zero temperature phase
diagram of metal-insulator and insulator-insulator transitions at half-filling
is mapped out which exhibits the interplay between $U$ and $g$. As $U$ ($g$) is
increased, a metal to Mott-Hubbard insulator (bipolaron insulator) transition
occurs, and the two insulating states are distinct and can not be adiabatically
connected. The nature of and transitions between the three states are
discussed.
