At temperatures above the weak phase transition, baryon and lepton number are badly violated. We explore the suggestion that the baryon asymmetry might be produced at the transition, if the transition is first order. We find that the asymmetry is proportional to a CP violating parameter, a large power of the gauge coupling and to factors which depend on the details of the transition. In extensions of the standard model, such as multi-Higgs or supersymmetric models, the result may be consistent with observations.