  A complete calculation of the finite temperature effective potential for the
abelian Higgs model to the order $e^4,\lambda^2$ is presented and the result is
expressed in terms of physical parameters defined at zero temperature. The
absence of a linear term is verified explicitly to the given order and proven
to survive to all orders. The first order phase transition has weakened in
comparison with lower order calculation, which shows up in a considerable
decrease of the surface tension. The only difference from the original version
is the splitting of some overlong lines causing problems with certain mailers.
