The interplay of gauge dynamics and flavor symmetries often leads to remarkably subtle phenomena in the presence of soliton configurations. Non-Abelian vortices-vortex solutions with continuous internal orientational moduli-provide an example. Here we study the effect of weakly gauging a U(1)(R) subgroup of the flavor symmetry on such BPS vortex solutions. Our prototypical setting consists of an SU(2) x U(1) gauge theory with N-f = 2 sets of fundamental scalars that break the gauge symmetry to an "electromagnetic" U(1). The weak U(1)(R) gauging converts the well-known CP1 orientation modulus vertical bar B vertical bar of the non-Abelian vortex into a parameter characterizing the strength of the magnetic field that is responsible for the Aharonov-Bohm effect. As the phase of B remains a genuine zero mode while the electromagnetic gauge symmetry is Higgsed in the interior of the vortex, these solutions are superconducting strings.