  The thermal conductivity $\kappa$ of underdoped \Y was measured in the $T \to
0$ limit as a function of hole concentration $p$ across the superconducting
critical point at $p_{SC}$ = 5.0%. ``Time doping'' was used to resolve the
evolution of bosonic and fermionic contributions with high accuracy. For $p
\leqslant p_{SC}$, we observe an additional $T^3$ contribution to $\kappa$
which we attribute to the boson excitations of a phase with long-range spin or
charge order. Fermionic transport, manifest as a linear term in $\kappa$, is
seen to persist unaltered through $p_{SC}$, showing that the state just below
$p_{SC}$ is a thermal metal. In this state, the electrical resistivity varies
as log$(1/T)$ and the Wiedemann-Franz law is violated.
