  The equation of state of the SU($3$) Yang-Mills theory is determined in the
deconfined phase with a precision of about 0.5%. The calculation is carried out
by numerical simulations of lattice gauge theory with shifted boundary
conditions in the time direction. At each given temperature, up to $230\, T_c$
with $T_c$ being the critical temperature, the entropy density is computed at
several lattice spacings so to be able to extrapolate the results to the
continuum limit with confidence. Taken at face value, above a few $T_c$ the
results exhibit a striking linear behaviour in $\ln(T/T_c)^{-1}$ over almost 2
orders of magnitude. Within errors, data point straight to the Stefan-Boltzmann
value but with a slope grossly different from the leading-order perturbative
prediction. The pressure is determined by integrating the entropy in the
temperature, while the energy density is extracted from $T s=(\epsilon + p )$.
The continuum values of the potentials are well represented by Pad\'e
interpolating formulas, which also connect them well to the Stefan-Boltzmann
values in the infinite temperature limit. The pressure, the energy and the
entropy densities are compared with results in the literature. The discrepancy
among previous computations near $T_c$ is analyzed and resolved thanks to the
high precision achieved.
