In *Hyperbolic groups* (page 82), Gromov claims that, by a standard application of Thurston's method of *geodesic (hyperbolic) simplices*, it can be prove that a hyperbolic group contains finitely many pairwise non conjugate subgroups isomorphic to a fixed one-ended finitely presented group. 


I took a look on the usual references dealing with hyperbolic groups, but I didn't find any details. Do you know references about this result and/or Thurston's method?



One-ended finitely presented subgroups of hyperbolic groups
There is a proof of this theorem due to Thomas Delzant:


T. Delzant, L’image d’un groupe dans un groupe hyperbolique, Comment. Math. Helv.
70 (1995), no. 2, 267–284.


There is also a version for relatively hyperbolic groups du to Francois Dahmani:


Accidental Parabolics and Relatively Hyperbolic Groups, Israel Journal of Mathematics, Volume 153, Issue 1, pp 93–127


