  We consider the one-dimensional focusing nonlinear Schr\"odinger equation
(NLS) with a delta potential and even initial data. The problem is equivalent
to the solution of the initial/boundary problem for NLS on a half-line with
Robin boundary conditions at the origin. We follow the method of Bikbaev and
Tarasov which utilizes a B\"acklund transformation to extend the solution on
the half-line to a solution of the NLS equation on the whole line. We study the
asymptotic stability of the stationary 1-soliton solution of the equation under
perturbation by applying the nonlinear steepest-descent method for
Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens,
and extends, earlier work on the problem by Holmer and Zworski.
