We report low-temperature thermal conductivity measurements on high-quality single crystalline α-BiPd, a noncentrosymmetric superconductor with Tc ≃ 3.8 K. In zero magnetic field, the residual linear term κ0/T is absent, indicating a full gap and suggesting dominant singlet pairing. Consistent with this picture, κ0(H)/T shows weak field dependence in low field, but grows relatively fast in fields above 0.03 T (≃1/5Hc2). Implications for the material's disputed high-field behavior are discussed, and constraints are placed on the previously proposed scenarios.