  We have determined the frequency shift that blackbody radiation is inducing
on the $5s^2$ $^1$S$_0$ -- $5s5p$ $^3$P$_0$ clock transition in strontium.
Previously its uncertainty limited the uncertainty of strontium lattice clocks
to $1\times10^{-16}$. Now the uncertainty associated to the black body
radiation shift correction translates to $5\times 10^{-18}$ relative frequency
uncertainty at room temperature. Our evaluation is based on a measurement of
the differential dc-polarizability of the two clock states and on a modeling of
the dynamic contribution using this value and experimental data for other
atomic properties.
