Asymptotic stability of small bound states in one dimension is proved in the framework of a discrete nonlinear Schrödinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenva lue. The analysis relies on the dispersive decay estimates from Pelinovsky and Stefanov [J. Math. Phys., 49 (2008), 113501] and the arguments of Mizumachi [J. Math. Kyoto Univ., 48 (2008), pp. 471-497] for a continuous nonlinear Schrödinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable bound states is higher than the one used in the analysis. © 2009 Society for Industrial and Applied Mathematics.