  We prove that the 1D Schr\"odinger equation with derivative in the nonlinear
term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The
result follows from an application of the ``I-method''. This method allows to
define a modification of the energy norm $H^{1}$ that is ``almost conserved''
and can be used to perform an iteration argument. We also remark that the same
argument can be used to prove that any quintic nonlinear defocusing
Schr\"odinger equation on the line is globally well-posed for large data in
$H^{s}$, for $s>2/3$ .
