The phase diagram of the Ising model with antiferromagnetic nearest-neighbor interactions and ferromagnetic equivalent-neighbor interactions on the Cayley tree is determined exactly. A nonuniversal critical line separates the disordered and the ordered phases. A line of first-order transitions separating ferromagnetic order from antiferromagnetic order ends in the midst of the ordered phase at a classical ordered critical point. For a small range of values of the ratio of the two couplings, two transitions occur as the temperature is varied. In this case the uniform magnetization is not a monotonic function of the temperature.