Consider a system of particles performing nearest neighbor random walks on the lattice $ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $a$--stable law, $0<a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/alpha}L(N)$, the density profile evolves as the solution of the random equation $partial_t rho = mf L_W rho$, where $mf L_W$ is the generalized second-order differential operator $frac d{du} frac d{dW}$ in which $W$ is a double sided $a$--stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array ${xi_{N,x} : xinbb Z}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.
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