We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).