  We study small charged black holes in global AdS spacetime in the presence of
a charged massless minimally coupled scalar field. In a certain parameter range
these black holes suffer from well known superradiant instabilities. We
demonstrate that the end point of the resultant tachyon condensation process is
a hairy black hole which we construct analytically in a perturbative expansion
in the black hole radius. At leading order our solution is a small undeformed
RNAdS black hole immersed into a charged scalar condensate that fills the AdS
`box'. These hairy black hole solutions appear in a two parameter family
labelled by their mass and charge. Their mass is bounded from below by a
function of their charge; at the lower bound a hairy black hole reduces to a
regular horizon free soliton which can also be thought of as a nonlinear Bose
condensate. We compute the microcanonical phase diagram of our system at small
mass, and demonstrate that it exhibits a second order `phase transition'
between the RNAdS black hole and the hairy black hole phases.
