Tight-binding calculations predict that the AA-stacked graphene bilayer has one electron and one hole conducting bands, and that the Fermi surfaces of these bands coincide. We demonstrate that as a result of this degeneracy, the bilayer becomes unstable with respect to a set of spontaneous symmetry violations. Which of the symmetries is broken depends on the microscopic details of the system. We find that antiferromagnetism is the more stable order parameter. This order is stabilized by the

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... ng on-site Coulomb repulsion. For an on-site repulsion energy typical for graphene systems, the antiferromagnetic gap can exist up to room temperatures.