  Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The
deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of
the corresponding Riemann surface. In genus one, the flat metric of the
deformed Frobenius manifold coincides with a metric associated with a
one-parameter family of solutions to the Painlev\'e-VI equation with
coefficients $(1/8,-1/8,1/8,3/8).$ Analogous deformations of the real doubles
of the Hurwitz Frobenius manifolds are also found; these deformations depend on
$g(g+1)/2$ real parameters.
