Let $M$ be a torus bundle over $S^1$ with an orientation preserving Anosov monodromy. The manifold $M$ admits a geometric structure modeled on Sol. We prove that the $\Sol$ structure can be deformed into singular hyperbolic cone structures whose singular locus $\Sigma\subset M$ is the mapping torus of the fixed point of the monodromy.\\ \hspace*{3mm} The hyperbolic cone metrics are parametred by the cone angle $\alpha$ in the interval $(0,2\pi)$. When $\alpha\to 2\pi$, the cone manifolds collapse to the basis of the fibration $S^1$, and they can be rescaled in the direction of the fibers to converge to the Sol manifold.