  In this article we continue the study of the quenched distributions of
transient, one-dimensional random walks in a random environment. In a previous
article we showed that while the quenched distributions of the hitting times do
not converge to any deterministic distribution, they do have a weak weak limit
in the sense that - viewed as random elements of the space of probability
measures - they converge in distribution to a certain random probability
measure (we refer to this as a weak weak limit because it is a weak limit in
the weak topology). Here, we improve this result to the path-valued process of
hitting times. As a consequence, we are able to also prove a weak weak quenched
limit theorem for the path of the random walk itself.
