It is shown that the SU(2) semilocal model -- the Abelian Higgs model with two complex scalars -- admits a new class of stationary, straight string solutions carrying a persistent current and having finite energy per unit length. In the plane orthogonal to their direction they correspond to a nontrivial deformation of the embedded Abrikosov-Nielsen-Olesen (ANO) vortices by the current flowing through them. The new solutions bifurcate with the ANO vortices in the limit of vanishing current. They

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... can be either static or stationary. In the stationary case the relative phase of the two scalars rotates at a constant velocity, giving rise to an electric field and angular momentum, while the energy remains finite. The current has a strong localizing effect on the magnetic field, thus evading the known spreading instability of the ANO-semilocal vortex solutions. The new static vortex solutions have lower energy than the ANO vortices and could be of considerable importance in various physical systems (from condensed matter to cosmic strings).