  We study a class of time dependent solutions of the vacuum Einstein equations
which are plane waves with weak null singularities. This singularity is weak in
the sense that though the tidal forces diverge at the singularity, the rate of
divergence is such that the distortion suffered by a freely falling observer
remains finite. Among such weak singular plane waves there is a sub-class which
do not exhibit large back reaction in the presence of test scalar probes.
String propagation in these backgrounds is smooth and there is a natural way to
continue the metric beyond the singularity. This continued metric admits string
propagation without the string becoming infinitely excited. We construct a one
parameter family of smooth metrics which are at a finite distance in the space
of metrics from the extended metric and a well defined operator in the string
sigma model which resolves the singularity.
