  We consider compactifications of the heterotic string on $K3 \times T2$ so
that the resulting theory in $d = 4$ space-time dimensions has $N = 2$
supersymmetry. The gravitational and gauge coupling constants of the low-energy
effective theory receive threshold corrections from loops of super-heavy string
states. We calculate these corrections for the case when the $K3$-surface is a
${\bf Z}_n$ orbifold of a four torus $T4$. The results are used to determine
the one-loop prepotential ${\cal F}_0^{(1)}$ for the vector multiplets and the
gravitational coupling ${\cal F}_1^{(1)}$.
