  We report low-temperature thermal conductivity down to 40 mK of the
antiferromagnet BaFe$_2$As$_{2}$, which is the parent compound of recently
discovered iron-based superconductors. In the investigated temperature range
below 4 K, the thermal conductivity $\kappa$ is well described by the
expression $\kappa$ = $aT$ + $bT^{2.22}$. We attribute the ``$aT$''-term to an
electronic contribution which is found to satisfy the Wiedemann-Franz law in
the $T$ $\to$ 0 K limit, and the remaining thermal conductivity, $\sim$
$T^{2.22}$, is attributed to phonon conductivity. A small influence on thermal
conductivity by magnetic fields up to 8 T is well accounted by the observed
magnetoresistance. The result is consistent with a fully gapped magnon
spectrum, inferred previously from inelastic neutron scattering measurements.
