We measured the ultra-low-temperature specific heat and thermal conductivity of the Au2Pb single crystal, a possible three-dimensional Dirac semimetal with a superconducting transition temperature $T_c \approx 1.05\ \text{K}$ . The electronic specific heat can be fitted by a two-band s-wave model, which gives the gap amplitudes Δ1(0)/kBTc = 1.41 and Δ2(0)/kBTc = 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 Au2Pb has a fully gapped superconducting state in the bulk, which is a necessary condition for topological superconductors if Au2Pb is indeed one.