  We show that a conformal connection on a closed oriented surface $\Sigma$ of
negative Euler characteristic preserves precisely one conformal structure and
is furthermore uniquely determined by its unparametrised geodesics. As a
corollary it follows that the unparametrised geodesics of a Riemannian metric
on $\Sigma$ determine the metric up to constant rescaling. It is also shown
that every conformal connection on the $2$-sphere lies in a complex
$5$-manifold of conformal connections, all of which share the same
unparametrised geodesics.
