  We consider the nonlinear Schr\"{o}dinger equation on the half-line $x \geq
0$ with a Robin boundary condition at $x = 0$ and with initial data in the
weighted Sobolev space $H^{1,1}(\mathbb{R}_+)$. We prove that there exists a
global weak solution of this initial-boundary value problem and provide a
representation for the solution in terms of the solution of a Riemann--Hilbert
problem. Using this representation, we obtain asymptotic formulas for the
long-time behavior of the solution. In particular, by restricting our
asymptotic result to solutions whose initial data are close to the initial
profile of the stationary one-soliton, we obtain results on the asymptotic
stability of the stationary one-solitons under any small perturbation in
$H^{1,1}(\mathbb{R}_+)$. In the focusing case, such a result was already
established by Deift and Park using different methods, and our work provides an
alternative approach to obtain such results.
