  We study the thermal conductivity of the one dimensional Toda lattice
perturbed by a stochastic dynamics preserving energy and momentum. The strength
of the stochastic noise is controlled by a parameter $\gamma$. We show that
heat transport is anomalous, and that the thermal conductivity diverges with
the length $n$ of the chain according to $\kappa(n) \sim n^\alpha$, with $0 <
\alpha \leq 1/2$. In particular, the ballistic heat conduction of the
unperturbed Toda chain is destroyed. Besides, the exponent $\alpha$ of the
divergence depends on $\gamma$.
