  In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger
equation (NLS) with bounded initial data which do not vanish at infinity. Local
well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove
global well-posedness for the 1D NLS with initial data lying in $L^{p}$ for any
$2 < p < \infty$, provided the initial data is sufficiently smooth. We do not
use the complete integrability of the cubic nonlinear Schr{\"o}dinger equation.
