  Let $S$ be a closed, orientable surface of genus at least 2. The cotangent
bundle of the "hyperbolic'' Teichm\"uller space of $S$ can be identified with
the space $\CP$ of complex projective structures on $S$ through measured
laminations, while the cotangent bundle of the "complex'' Teichm\"uller space
can be identified with $\CP$ through the Schwarzian derivative. We prove that
the resulting map between the two cotangent spaces, although not smooth, is
symplectic. The proof uses a variant of the renormalized volume defined for
hyperbolic ends.
