  Menasco showed that a non-split, prime, alternating link that is not a
2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed
thickened surfaces $S \times I$. We define a link to be fully alternating if it
has an alternating projection from $S\times I$ to $S$ where the interior of
every complementary region is an open disk. We show that a prime, fully
alternating link in $S\times I$ is hyperbolic. Similar to Menasco, we also give
an easy way to determine primeness in $S\times I$. A fully alternating link is
prime in $S\times I$ if and only if it is "obviously prime". Furthermore, we
extend our result to show that a prime link with fully alternating projection
to an essential surface embedded in an orientable, hyperbolic 3-manifold has a
hyperbolic complement.
