Classical solutions describing strings endowed with an electric charge and carrying a constant electromagnetic current are constructed within the bosonic sector of the Electroweak Theory. For any given ratio of the Higgs boson mass to W boson mass and for any Weinberg angle, these strings comprise a family that can be parameterized by values of the current through their cross-section, I3, by their electric charge per unit string length, I0, and by two integers. These parameters determine the electromagnetic and Z fluxes, as well as the angular momentum and momentum densities of the string. For I0 -> 0 and I3 -> 0 the solutions reduce to Z strings, or, for solutions with I0 = ± I3, to the W-dressed Z strings whose existence was discussed some time ago.