  The thermal conductivity $\kappa$ of the cuprate superconductor
La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ 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 $\kappa_0/T$ as $T \to 0$ across the pseudogap critical
point $p^{\star}= 0.23$. In the normal state, we observe an abrupt drop in
$\kappa_0/T$ upon crossing below $p^{\star}$, 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 $\kappa_0/T$ 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,
$\kappa_0/T=L_0/\rho(0)$, is obeyed at all dopings, including at the critical
point where the electrical resistivity $\rho(T)$ is $T$-linear down to $T \to
0$. We conclude that the non-superconducting ground state of the pseudogap
phase at $T=0$ is a metal whose fermionic excitations carry heat and charge as
conventional electrons do.
