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