  Let $S$ be a closed oriented surface of genus $g\geq 2$. Fix an arbitrary
non-elementary representation $\rho\colon\pi_1(S)\to {\rm SL}_2(\mathbb{C})$
and consider all marked (complex) projective structures on $S$ with holonomy
$\rho$. We show that their underlying conformal structures are dense in the
moduli space of $S$.
