  We prove that, for a natural class of Bridgeland stability conditions on
$\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point,
the moduli spaces of Bridgeland semistable objects are projective. Our
technique is to find suitable regions of stability conditions with hearts that
are (after "rotation") equivalent to representations of a quiver. The helix and
tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that
this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo
surfaces, and the analogous Bridgeland moduli spaces are projective.
