  We study the gravitational interaction involving the dilaton and the
anti-symmetrical $B_{\mu\nu}$ fields that arises in the low-energy limit of
string theory. It is shown that such interaction can be derived from a
geometrical action principle, with the scalar of curvature of a non-Riemannian,
but {\em metric-compatible}, connection as the lagrangian, and with a
non-parallel volume-element. This action is contrasted with the recently
proposed geometrical action for the 4-dimensional axi-dilaton gravity.
