A gauged (2+1)-dimensional version of the Skyrme model is investigated. The gauge group is U(1) and the dynamics of the associated gauge potential is governed by a Maxwell term. In this model there are topologically stable soliton solutions carrying magnetic flux which is not topologically quantized. The properties of static, rotationally symmetric solitons of degree one and two are discussed in detail. It is shown that the electric field of such solutions is necessarily zero. The solitons' shape, mass, and magnetic flux depend on the U(1) coupling constant, and this dependence is studied numerically from very weak to very strong coupling.