  Let \theta:\pi_1(R) \to \PSL(2,\C) be a homomorphism of the fundamental group
of an oriented, closed surface R of genus exceeding one. We will establish the
following theorem.
  Theorem. Necessary and sufficient for \theta to be the monodromy
representation associated with a complex projective stucture on R, either
unbranched or with a single branch point of order 2, is that \theta(\pi_1(R))
be nonelementary. A branch point is required if and only if the representation
\theta does not lift to \SL(2,\C).
