  We generalize the result of Lukacs {\it et al.} on asymptotic stability of
the Schwarzschild metric with respect to perturbations in the Robinson-Trautman
class of metrics to the case of Petrov type II twisting metrics, uder the
condition of asymptotic flatness at future null infinity. The Bondi energy is
used as the Lyapunov functional and we prove that the "final state" of such
metrics is the Kerr metric.
