  Black-body radiation (BBR) shift is an important systematic correction for
the atomic frequency standards realizing the SI unit of time. Presently, there
is a controversy over the value of the BBR shift for the primary $^{133}$Cs
standard. At room temperatures the values from various groups differ at $3
\times 10^{-15}$ level, while the modern clocks are aiming at $10^{-16}$
accuracies. We carry out high-precision relativistic many-body calculations of
the BBR shift. For the BBR coefficient $\beta$ at $T=300K$ we obtain
$\beta=-(1.708\pm0.006) \times 10^{-14}$, implying $6 \times 10^{-17}$
fractional uncertainty. While in accord with the most accurate measurement, our
0.35%-accurate value is in a substantial, 10%, disagreement with recent
semi-empirical calculations. We identify an oversight in those calculations.
