  We investigate analytically and numerically the asymptotic behavior of the
Nielsen-Olesen vortex solutions and show that they approach their asymptotic
values exponentially but with exponents that differ from the ones quoted in the
literature. In particular, it is shown that for values of the Higgs
self-coupling that are larger than a critical value, both the Higgs field and
the gauge field approach exponentially their asymptotic values with {\it equal}
exponents that are {\it independent} of the Higgs self-coupling.
