  We investigate whether baryogenesis is possible at a second order electroweak
phase transition. We find that under rather general conditions, the departure
from thermal equilibrium is suppressed by the expansion rate of the Universe,
and hence baryon production is also suppressed by the expansion rate. We
conclude that if no additional sources of departure from thermal equilibrium
such as topological defects are present, then electroweak baryogenesis is ruled
out if the phase transition is second order or crossover. However, a
non-vanishing net baryon to entropy ratio {\it is} generated, and we provide
both upper and lower bounds for the result. Our technique is also applicable to
other baryogenesis mechanisms taking place during and immediately after a
second order phase transition. We estimate the lowest value of the transition
energy scale for which the resulting baryon to entropy ratio might be large
enough to explain observations.
