  We consider a continuous-time random walk which is defined as an
interpolation of a random walk on a point process on the real line. The
distances between neighboring points of the point process are i.i.d. random
variables in the normal domain of attraction of an $\alpha$-stable distribution
with $0 < \alpha < 1$. This is therefore an example of a random walk in a
L\'evy random medium. Specifically, it is a generalization of a process known
in the physical literature as L\'evy-Lorentz gas. We prove that the annealed
version of the process is superdiffusive with scaling exponent $1/(\alpha + 1)$
and identify the limiting process, which is not c\`adl\`ag. The proofs are
based on the technique of Kesten and Spitzer for random walks in random
scenery.
