  We obtain a Lie theoretic intrinsic characterization of the connected and
simply connected solvable Lie groups whose regular representation is a factor
representation. When this is the case, the corresponding von Neumann algebras
are isomorphic to the hyperfinite II$_\infty$ factor, and every Casimir
function is constant. We thus obtain a family of geometric models for the
standard representation of that factor. Finally, we show that the regular
representation of any connected and simply connected solvable Lie group with
open coadjoint orbits is always of type I, though the group needs not be of
type I, and include some relevant examples.
