  Electroweak baryogenesis can occur in the ``adiabatic limit,'' in which
expanding bubbles of true vacuum are assumed to have rather thick walls and be
slowly moving. Here the problem of calculating the baryon asymmetry in this
limit is reconsidered.
  A simple prescription for obtaining the relevant kinetic equations is given.
An additional suppression beyond that usually assumed is found.
  This arises because the generation of an asymmetry requires violation of
approximately conserved currents by Higgs expectation values, which are small
near the front of the walls.
  As applications, the asymmetries in multi-Higgs models and the minimal
supersymmetric standard model are estimated to be proportional to $\alpha_w^8$
and $\alpha_w^6$ respectively.
  Also, in this limit the baryon asymmetry in the Minimal Standard Model is
extraordinarily small.
