  We show the existence of weak solutions in the extended sense of the Cauchy
problem for the cubic nonlinear Schr\"odinger equation in the modulation space
$M_{2,q}^{s}(\mathbb R)$, $1\leq q\leq2$ and $s\geq0.$ In addition, for either
$s\geq 0$ and $1\leq q\leq\frac32$ or $\frac32<q\leq 2$ and
$s>\frac23-\frac1{q}$ we show that the Cauchy problem is unconditionally
wellposed in $M_{2,q}^{s}(\mathbb R).$ It is done with the use of the
differentiation by parts technique which had been previously used in the
periodic setting.
