Dominant two-loop corrections to the MSSM finite temperature effective potential

We show that two-loop corrections to the finite temperature effective potential in the MSSM can have a dramatic effect on the strength of the electroweak phase transition, making it more strongly first order. The change in the order parameter v/T_c can be as large as 75\% of the one-loop daisy improved result. This effect can be decisive to widen the region in parameter space where erasure of the created baryons by sphaleron processes after the transition is suppressed and hence, where electroweak baryogenesis might be successful. We find an allowed region with \tan\beta\simlt 4.5 and a Higgs boson with standard couplings and mass below 80\ GeV within the reach of LEP II