  We study the decay of the metastable symmetric phase in the standard model at
finite temperature. For the SU(2)-Higgs model the two wave function correction
terms $Z_{\vp}(\vp^2,T)$ and $Z_{\chi}(\vp^2,T)$ of Higgs and Goldstone boson
fields are calculated to one-loop order. We find that the derivative expansion
of the effective action is reliable for Higgs masses smaller than the W-boson
mass. We propose a new procedure to evaluate the decay rate by first
integrating out the vector field and the components of the scalar fields with
non-zero Matsubara frequencies. The static part of the scalar field is treated
in the saddle point approximation. As a by-product we obtain a formula for the
decay rate of a homogeneous unstable state. The course of the cosmological
electroweak phase transition is evaluated numerically for different Higgs boson
masses and non-vanishing magnetic mass of the gauge boson. For Higgs masses
above $\sim 60$ GeV the latent heat can reheat the system to the critical
temperature which qualitatively changes the nature of the transition.
