We consider an Abelian Gauge Theory in R⁴ equipped with the Minkowski metric. This theory leads to a system of equations, the Klein-Gordon-Maxwell equations, which provide models for the interaction between the electromagnetic field and matter. We assume that the nonlinear term is such that the energy functional is positive; this fact makes the theory more suitable for physical models. A three dimensional vortex is a finite energy, stationary solution of these equations such that the matter field has nontrivial angular momentum and the magnetic field looks like the field created by a finite solenoid. Under suitable assumptions, we prove the existence of three dimensional vortex-solutions.