We analyze the stability of global O(3) monopoles in the infinite cutoff (or scalar mass) limit. We obtain the perturbation equations and prove that the spherically symmetric solution is classically stable (or neutrally stable) to axially symmetric, square integrable, or power-law decay perturbations. Moreover, we show that, in spite of the existence of a conserved topological charge, the energy barrier between the monopole and the vacuum is finite even in the limit where the cutoff is taken to infinity. This feature is specific of global monopoles and independent of the details of the scalar potential.