The time evolution of the baryon asymmetry of the universe due to superheavy Higgs bosons is obtained by integrating the Boltzmann equations. The interactions included are decays, inverse decays, and annihilations of the Higgs bosons. The minimal SU(5) model is used to calculate the interactions, although our results are more generally applicable. Decays and inverse decays of these bosons damp preexisting asymmetries by exp(-0.3K) to exp(-2.0K) where K=2.9×1017αHGeVMH (αH and MH are the coupling strength and mass of the Higgs boson). If both C and CP are violated, then in an initially symmetrical universe a baryon asymmetry evolves and its value depends upon K and α2αH (α is the gauge coupling strength). For KαHα2 the asymmetry produced is knBs2×10-3ε, and for K1 it is knBs2×10-3ε(3K)-1.2 (ε2=the baryon excess produced by the decay of a pair of H, H̄ bosons). In SU(5) for MH3×1013 GeV preexisting asymmetries are not damped and the observed asymmetry knBs=10-9.8±1.6 can be produced if ε10-7. In a companion paper the role of gauge bosons is considered.