  We show that gated bilayer graphene hosts a strong topological insulator (TI)
phase in the presence of Rashba spin-orbit (SO) coupling. We find that gated
bilayer graphene under preserved time-reversal symmetry is a quantum valley
Hall insulator for small Rashba SO coupling $\lambda_{\mathrm{R}}$, and
transitions to a strong TI when $\lambda_{\mathrm{R}} > \sqrt{U^2+t_\bot^2}$,
where $U$ and $t_\bot$ are respectively the interlayer potential and tunneling
energy. Different from a conventional quantum spin Hall state, the edge modes
of our strong TI phase exhibit both spin and valley filtering, and thus share
the properties of both quantum spin Hall and quantum valley Hall insulators.
The strong TI phase remains robust in the presence of weak graphene intrinsic
SO coupling.
