The thermal conductivity κ of the cuprate superconductor La1.6-xNd0.4SrxCuO4 was measured down to 50 mK in seven crystals with doping from p=0.12 to p=0.24, both in the superconducting state and in the magnetic field-induced normal state. We obtain the electronic residual linear term κ0/T as T→0 across the pseudogap critical point p=0.23. In the normal state, we observe an abrupt drop in κ0/T upon crossing below p, consistent with a drop in carrier density n from 1+p to p, the signature of the pseudogap phase inferred from the Hall coefficient. A similar drop in κ0/T is observed at H=0, showing that the pseudogap critical point and its signatures are unaffected by the magnetic field. In the normal state, the Wiedemann-Franz law, κ0/T=L0/ρ(0), is obeyed at all dopings, including at the critical point where the electrical resistivity ρ(T) is T linear down to T→0. We conclude that the nonsuperconducting ground state of the pseudogap phase at T=0 is a metal whose fermionic excitations carry heat and charge as conventional electrons do.