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An anomaly-free U(1) gauge extension of the standard model (SM) is presented. Only one Higgs doublet with a nonzero vacuum expectation is required as in the SM. New fermions and scalars as well as all SM particles transform nontrivially under this U(1), resulting in a model of three active neutrinos and one sterile neutrino, all acquiring radiative masses. Charged-lepton masses are also radiative as well as the mixing between active and sterile neutrinos. At the same time, a residual Z2 symmetry of the U(1) gauge symmetry remains exact, allowing for the existence of dark matter.