  We study the thermal conductivity, at fixed positive temperature, of a
disordered lattice of harmonic oscillators, weakly coupled to each other
through anharmonic potentials. The interaction is controlled by a small
parameter $\epsilon > 0$. We rigorously show, in two slightly different setups,
that the conductivity has a non-perturbative origin. This means that it decays
to zero faster than any polynomial in $\epsilon$ as $\epsilon\rightarrow 0$. It
is then argued that this result extends to a disordered chain studied by Dhar
and Lebowitz, and to a classical spins chain recently investigated by
Oganesyan, Pal and Huse.
