  We reconsider the strength of the electroweak phase transition in the singlet
Majoron extension of the Standard Model, with a low (~TeV) scale of the singlet
VEV. A strongly first order phase transition, of interest for electroweak
baryogenesis, is found in sizeable regions of the parameter space, especially
when the cross-coupling lambda_{hs}|S|^2|H|^2 between the singlet and the
doublet Higgs is significant. Large Majorana Yukawa couplings of the singlet
neutrinos, y_i S nu_i^c nu_i, are also important for strengthening the
transition. We incorporate the LEP and Tevatron constraints on the Higgs
masses, and electroweak precision constraints, in our search for allowed
parameters; successful examples include singlet masses ranging from 5 GeV to
several TeV. Models with a strong phase transition typically predict a
nonstandard Higgs with mass in the range 113 GeV < m_H < 200 GeV and production
cross sections reduced by mixing with the singlet, with cos^2(theta)
significantly less than 1. We also find examples where the singlet is light and
the decay H -> SS can modify the Higgs branching ratios relative to Standard
Model expectations.
