  We consider a one-dimensional, transient random walk in a random i.i.d.
environment. The asymptotic behaviour of such random walk depends to a large
extent on a crucial parameter $\kappa>0$ that determines the fluctuations of
the process. When $0<\kappa<2$, the averaged distributions of the hitting times
of the random walk converge to a $\kappa$-stable distribution. However, it was
shown recently that in this case there does not exist a quenched limiting
distribution of the hitting times. That is, it is not true that for almost
every fixed environment, the distributions of the hitting times (centered and
scaled in any manner) converge to a non-degenerate distribution. We show,
however, that the quenched distributions do have a limit in the weak sense.
That is, the quenched distributions of the hitting times -- viewed as a random
probability measure -- converge in distribution to a random probability
measure, which has interesting stability properties. Our results generalize
both the averaged limiting distribution and the non-existence of quenched
limiting distributions.
