  With the physical Higgs mass the Standard Model symmetry restoration phase
transition is a smooth cross-over. We study the thermodynamics of the
cross-over using numerical lattice Monte Carlo simulations of an effective
SU(2) X U(1) gauge + Higgs theory, significantly improving on previously
published results. We measure the Higgs field expectation value, thermodynamic
quantities like pressure, energy density, speed of sound and heat capacity, and
screening masses associated with the Higgs and Z fields. While the cross-over
is smooth, it is very well defined with a width of only approximately 5 GeV. We
measure the cross-over temperature from the maximum of the susceptibility of
the Higgs condensate, with the result $T_c = 159.5 \pm 1.5$ GeV. Outside of the
narrow cross-over region the perturbative results agree well with
non-perturbative ones.
