  We give a complete characterization of the locally compact groups that are
non-elementary Gromov-hyperbolic and amenable. They coincide with the class of
mapping tori of discrete or continuous one-parameter groups of compacting
automorphisms. We moreover give a description of all Gromov-hyperbolic locally
compact groups with a cocompact amenable subgroup: modulo a compact normal
subgroup, these turn out to be either rank one simple Lie groups, or
automorphism groups of semi-regular trees acting doubly transitively on the set
of ends. As an application, we show that the class of hyperbolic locally
compact groups with a cusp-uniform non-uniform lattice, is very restricted.
