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We study magnetically charged black holes in the Einstein-Yang-Mills-Higgs theory in the limit of infinitely strong coupling of the Higgs field. Using mixed analytical and numerical methods we give a complete description of static spherically symmetric black hole solutions, both Abelian and non-Abelian. In particular, we find a new class of extremal non-Abelian solutions. \Ve show that all non-Abelian solutions are stable against linear radial perturbations. The implications of our results for the semiclassical evolution of magnetically charged black holes are discussed. PACS number(s): 04.20Jb, 11.15.Kc, 97.60Lf

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