  We use Heegaard decompositions and the theta divisor on a Riemannian surface
to define a three-manifold invariant for rational homology three-spheres. This
invariant is defined on the set of $Spin^c$ structures $$ {\hat \theta}\colon
Spin^c(Y)\longrightarrow Q. $$ In the first part of the paper, we give the
definition of the invariant (which builds on the theory developed in the
article, "The theta divisor and three-manifold invariants"). In the second
part, we establish a relationship between this invariant and the Casson-Walker
invariant.
