  We propose and study a conformal field theory (CFT) model with random
position-dependent velocity that, as we argue, naturally emerges as an
effective description of heat transport in one-dimensional quantum many-body
systems with certain static random impurities. We present exact analytical
results that elucidate how purely ballistic heat waves in standard CFT can
acquire normal and anomalous diffusive contributions due to our impurities. Our
results include impurity-averaged Green's functions describing the time
evolution of the energy density and the heat current, and an explicit formula
for the thermal conductivity that, in addition to a universal Drude peak, has a
nontrivial real regular contribution that depends on details of the impurities.
