A gauge theory with U(N) symmetry is studied in the large-N limit to all orders in the scalar self-coupling, and to lowest nontrivial order in the gauge coupling. Unlike the 1N expansion of the scalar field theory, the effective potential is found to be always real, although it can be mutiple-valued. Furthermore, there is a region in the coupling-constant space where symmetry is broken spontaneously involving two separate phase transitions, one of the first kind and one of the second kind. This phenomenon persists for arbitrarily small (finite) gauge coupling as a genuine feature of the 1N expansion, exhibiting much more of the nonlinear structure of the complete theory than found in ordinary perturbation expansions. A comparison is made between the spontaneous symmetry breaking found in the 1N expansion and that of other symmetry-breaking schemes, which are of the Goldstone, Higgs, or Coleman-Weinberg type. Here the vector-scalar-boson mass ratio, MV2MS2, is of O(g2), which is contrasted with Higgs and Coleman-Weinberg mechanisms, for which MV2MS2 is of O(1) and O(g-2), respectively, where g is the gauge coupling.