  We use the theory of group actions on profinite trees to prove that the
fundamental group of a finite, 1-acylindrical graph of free groups with
finitely generated edge groups is conjugacy separable. This has several
applications: we prove that positive, $C'(1/6)$ one-relator groups are
conjugacy separable; we provide a conjugacy separable version of the Rips
construction; we use this latter to provide an example of two finitely
presented, residually finite groups that have isomorphic profinite completions,
such that one is conjugacy separable and the other does not even have solvable
conjugacy problem.
