  The invariant measured foliations of a pseudo-Anosov homeomorphism induce a
natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov
monodromy. We show that when the pseudo-Anosov $\phi:S\rightarrow S$ has
orientable foliations and does not have 1 as an eigenvalue of the induced
cohomology action on the closed surface, then the Sol structure can be deformed
to nearby cone hyperbolic structures, in the sense of projective structures.
The cone angles can be chosen to be decreasing from multiples of $2\pi$.
