  As another application of the degeneration methods of [V3], we count the
number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed
multiple points on a conic $E$, not containing $E$, through an appropriate
number of general points in the plane. As a special case, we count the number
of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the
plane at up to five points (no three collinear). We then show that these
numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we
suggest a direction from which the remaining del Pezzo surfaces can be
approached, and give a conjectural algorithm to compute the genus g
Gromov-Witten invariants of the cubic surface.
