  Using a Hamiltonian approach to gauged WZW models, we present a general
method for computing the conformally exact metric and dilaton, to all orders in
the $1/k$ expansion, for any bosonic, heterotic, or type-II superstring model
based on a coset $G/H$. We prove the following relations: (i) For type-II
superstrings the conformally exact metric and dilaton are identical to those of
the non-supersymmetric {\it semi-classical} bosonic model except for an overall
renormalization of the metric obtained by $k\to k- g$. (ii) The exact
expressions for the heterotic superstring are derived from their exact bosonic
string counterparts by shifting the central extension $k\to 2k-h$ (but an
overall factor $(k-g)$ remains unshifted). (iii) The combination
$e^\Phi\sqrt{-G}$ is independent of $k$ and therefore can be computed in lowest
order perturbation theory as required by the correct formulation of a
conformally invariant path integral measure. The general formalism is applied
to the coset models $SO(d-1,2)_{-k}/SO(d-1,1)_{-k}$ that are relevant for
string theory on curved spacetime. Explicit expressions for the conformally
exact metric and dilaton for the cases $d=2,3,4$ are given. In the
semiclassical limit $(k\to \infty)$ our results agree with those obtained with
the Lagrangian method up to 1-loop in perturbation theory.
