We study the interactions of quarks and antiquarks with the changing Higgs field during the electroweak phase transition, including quantum mechanical and some thermal effects, with the only source of CP violation being the known CKM phase. We show that the GIM cancellation, which has been commonly thought to imply a prediction which is at least 10 orders of magnitude too small, can be evaded in certain kinematic regimes, for instance, when the strange quark is totally reflected but the down quark is not. We report on a quantitative calculation of the asymmetry in a one-dimensional approximation based on the present understanding of the physics of the high-temperature environment, but with some aspects of the problem oversimplified. The resulting prediction for the magnitude and sign of the present baryonic asymmetry of the Universe agrees with the observed value, with moderately optimistic assumptions about the dynamics of the phase transition. Both magnitude and sign of the asymmetry have an intricate dependence on quark masses and mixings, so that quantitative agreement between prediction and observation would be highly nontrivial. At present uncertainties related to the dynamics of the EW phase transition and the oversimplifications of our treatment are too great to decide whether or not this is the correct explanation for the presence of remnant matter in our Universe; however, the present work makes it clear that the minimal standard model cannot be discounted as a contender for explaining this phenomenon. © 1994 The American Physical Society.