  We calculate the dissipation of the baryon number after the electroweak phase
transition due to thermal fluctuations above the sphaleron barrier. We consider
not only the classical Boltzmann factor but also fermionic and bosonic one-loop
contributions. We find that both bosonic and especially fermionic fluctuations
can considerably suppress the transition rate. Assuming the Langer--Affleck
formalism for this rate, the condition that an initial baryon asymmetry must
not be washed out by sphaleron transitions leads, in the Minimal Standard Model
($\sin\theta_W=0$), to an upper bound for the Higgs mass in the range 60 to 75
GeV.
