  Let $G$ be the fundamental group of a sapphire that admits the Sol geometry
and is not a torus bundle. We determine a finite free resolution of
$\mathbb{Z}$ over $\mathbb{Z}G$ and calculate a partial diagonal approximation
for this resolution. We also compute the cohomology rings $H^*(G; A)$ for $A =
\mathbb{Z}$ and $A = \mathbb{Z}/p$ for an odd prime $p$, and indicate how to
compute the groups $H^*(G; A)$ and the multiplicative structure given by the
cup product for any system of coefficients $A$.
