A spherical black-hole global-monopole system in thermal equilibrium is examined from the perspective of a grand canonical ensemble in which the temperature and surface area are fixed at a finite boundary. Using the Euclidean action with constraints eliminated, we investigate the thermodynamics of the system. It is shown that there exists a stable thermal equilibrium for the system in the ensemble. A comparison with ordinary black holes under equivalent boundary conditions indicates that if such a system exists, it will not decay spontaneously to a black hole without a global monopole.