  We measured the ultra-low-temperature specific heat and thermal conductivity
of Au$_2$Pb single crystal, a possible three-dimensional Dirac semimetal with a
superconducting transition temperature $T_c \approx$ 1.05 K. The electronic
specific heat can be fitted by a two-band s-wave model, which gives the gap
amplitudes $\Delta_1$(0)/$k_BT_c$ = 1.38 and $\Delta_2$(0)/$k_BT_c$ = 5.25.
From the thermal conductivity measurements, a negligible residual linear term
$\kappa_0/T$ in zero field and a slow field dependence of $\kappa_0/T$ at low
field are obtained. These results suggest that Au$_2$Pb has a fully gapped
superconducting state in the bulk, which is a necessary condition for
topological superconductor if Au$_2$Pb is indeed one.
