  We measure the dynamic differential scalar polarizabilities at 10.6 $\mu$m
for two candidate clock transitions in $^{176}\mathrm{Lu}^+$. The fractional
black body radiation (BBR) shifts at 300 K for the $^1S_0 \leftrightarrow
{^3D_1}$ and $^1S_0 \leftrightarrow {^3D_2}$ transitions are evaluated to be
$-1.36\,(9) \times 10^{-18}$ and $2.70 \,(21) \times10^{-17}$, respectively.
The former is the lowest of any established optical atomic clock.
