  A well known question of Gromov asks whether every one-ended hyperbolic group
$\Gamma$ has a surface subgroup. We give a positive answer when $\Gamma$ is the
fundamental group of a graph of free groups with cyclic edge groups. As a
result, Gromov's question is reduced (modulo a technical assumption on
2-torsion) to the case when $\Gamma$ is rigid. We also find surface subgroups
in limit groups. It follows that a limit group with the same profinite
completion as a free group must in fact be free, which answers a question of
Remeslennikov in this case.
