  We consider a random walk among a Poisson system of moving traps on ${\mathbb
Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities
of this random walk have been investigated. Here we study the path of the
random walk conditioned on survival up to time $t$ in the annealed case and
show that it is subdiffusive. As a by-product, we obtain an upper bound on the
number of so-called thin points of a one-dimensional random walk, as well as a
bound on the total volume of the holes in the random walk's range.
