Is There a Hot Electroweak Phase Transition at $m_Hgsim m_W$?

We provide non-perturbative evidence for the fact that there is no hot electroweak phase transition at large Higgs masses, $m_H = 95$, 120 and 180 GeV. This means that the line of first order phase transitions separating the symmetric and broken phases at small $m_H$ has an end point $m_{H,c}$. In the minimal standard electroweak theory 70 GeV $<m_{H,c}<$ 95 GeV and most likely $m_{H,c} approx 80$ GeV. If the electroweak theory is weakly coupled and the Higgs boson is found to be heavier than the critical value (which depends on the theory in question), cosmological remnants from the electroweak epoch are improbable.