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We present an exact numerical computation of the one-loop correction of thestring tension for the Nielsen-Olesen vortex in the Abelian Higgs model. Thecomputations proceeds via the computation of the Euclidean Green's function forthe gauge, Higgs and Faddeev-Popov fields using mode functions, and taking theappropriate trace. Renormalization is an essential part of this computation, itis done by removing leading order contributions from the numerical results soas to make these finite, and to add the divergent parts back, after suitableregularization and renormalization. We encounter and solve some problems whichare specific to gauge theories and topological solutions. The corrections tothe energy are found to be sizeable, but still smaller than the classicalenergy as long as g**2 is smaller than unity.

One-loop corrections to the string tension of the vortex in the Abelian Higgs model