  Solutions to the Cauchy problem for the one-dimensional cubic nonlinear
Schr\"odinger equation on the real line are studied in Sobolev spaces $H^s$,
for $s$ negative but close to 0. For smooth solutions there is an {\em a
priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$
norm of the datum, for arbitrarily large data, for sufficiently short time.
Weak solutions are constructed for arbitrary initial data in $H^s$.
