The continuum 3D SU(2) × U(1)+Higgs theory is an effective theory for a large class of 4D high-temperature gauge theories, including the minimal standard model and some of its supersymmetric extensions. We study the effects of the U(1) subgroup using lattice Monte Carlo techniques. When g' 2/ g2 is increased from the zero corresponding to pure SU(2)+Higgs theory, the phase transition gets stronger. However, the increase in the strength is close to what is expected perturbatively, and the qualitative features of the phase diagram remain the same as for g' 2 = 0. In particular, the first-order transition still disappears for mH > mH, c. We measure the photon mass and mixing angle, and find that the mass vanishes in both phases within the statistical errors.