  We investigate the instability of the massive scalar field in the vicinity of
a rotating black hole. The instability arises from amplification caused by the
classical superradiance effect. The instability affects bound states: solutions
to the massive Klein-Gordon equation which tend to zero at infinity. We
calculate the spectrum of bound state frequencies on the Kerr background using
a continued fraction method, adapted from studies of quasinormal modes. We
demonstrate that the instability is most significant for the $l = 1$, $m = 1$
state, for $M \mu \lesssim 0.5$. For a fast rotating hole ($a = 0.99$) we find
a maximum growth rate of $\tau^{-1} \approx 1.5 \times 10^{-7} (GM/c^3)^{-1}$,
at $M \mu \approx 0.42$. The physical implications are discussed.
