  We obtain necessary conditions for the existence of special K\"ahler
structures with isolated singularities on compact Riemann surfaces. We prove
that these conditions are also sufficient in the case of the Riemann sphere
and, moreover, we determine the whole moduli space of special K\"ahler
structures with fixed singularities. The tool we develop for this aim is a
correspondence between special K\"ahler structures and pairs consisting of a
cubic differential and a hyperbolic metric.
