  Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free
nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let
$G$ be a compactly generated locally compact second countable group such that
the closure of the group of inner automorphisms of $G$ is compact in the
natural topology. It is shown that there exists a {\it bounded} ergodic
$G$-valued cocycle of $\Gamma$.
