  Scenarios for electroweak baryogenesis require an understanding of the
effective potential at finite temperature near a first-order electroweak phase
transition. Working in Landau gauge, we present a calculation of the dominant
two-loop corrections to the ring-improved one-loop potential in the formal
limit $g^4 \ll \lambda \ll g^2$, where $\lambda$ is the Higgs self-coupling and
$g$ is the electroweak coupling. The limit $\lambda \ll g^2$ ensures that the
phase transition is significantly first-order, and the limit $g^4 \ll \lambda$
allows us to use high-temperature expansions. We find corrections from 20 to
40\% at Higgs masses relevant to the bound computed for baryogenesis in the
Minimal Standard Model. Though our numerical results seem to still rule out
Minimal Standard Model baryogenesis, this conclusion is not airtight because
the loop expansion is only marginal when corrections are as big as 40\%. We
also discuss why super-daisy approximations do not correctly compute these
corrections.
