  In this paper we investigate the unitarity of gauged non-compact WZNW strings
i.e. string theories formulated as G/H' WZNW models, where G is a non-compact
group. These models represent string theories on non-trivial curved space-times
with one time-like component. We will prove that for the class of models
connected to Hermitian symmetric spaces, and a natural set of discrete highest
weight representations, the BRST formulation, in which the gauging is defined
through a BRST condition, yields unitarity. Unitarity requires the level times
the Dynkin index to be an integer, as well as integer valued highest weights
w.r.t. the compact subalgebra. We will also show that the BRST formulation is
not equivalent to the conventional GKO coset formulation, defined by imposing a
highest weight condition w.r.t. H'. The latter leads to non-unitary physical
string states.This is, to our knowledge, the first example of a fundamental
difference between the two formulations.
