  For any cosmological constant $\Lambda=-3/\ell^2<0$ and any $\alpha<9/4$, we
find a Kerr-AdS spacetime $(\mathcal M,g_{\mathrm{KAdS}})$, in which the
Klein-Gordon equation $\Box_{g_{\mathrm{KAdS}}}\psi+\alpha/\ell^2\psi=0$ has an
exponentially growing mode solution satisfying a Dirichlet boundary condition
at infinity. The spacetime violates the Hawking-Reall bound $r_+^2>|a|\ell$. We
obtain an analogous result for Neumann boundary conditions if $5/4<\alpha<9/4$.
Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime
violating the Hawking-Reall bound, there exists an open family of masses
$\alpha$ such that the corresponding Klein-Gordon equation permits
exponentially growing mode solutions. Our result adopts methods of
Shlapentokh-Rothman (see arXiv:1302.3448) and provides the first rigorous
construction of a superradiant instability for negative cosmological constant.
