We numerically construct asymptotically global AdS_5 x S^5 black holes that are localised on the S^5. These are solutions to type IIB supergravity with S^8 horizon topology that dominate the microcanonical ensemble at small energies. At higher energies, there is a first-order phase transition to AdS_5-Schwarzschild x S^5. By the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, this transition is dual to spontaneously breaking the SO(6) R-symmetry of N=4 supersymmetric Yang-Mills down to SO(5). We extrapolate the location of this phase transition and compute the expectation value of a scalar operator in the low energy phase. In addition, we discuss the construction of localised black holes in type IIA, which are dual (via T-duality) to the low temperature phase of thermal 1+1 dimensional supersymmetric Yang-Mills theory on a circle.