We obtain new results on the stability of discrete dark solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schrödinger equation, following the analysis of our previous paper (2005 Physica D 212 1–19). We derive a criterion for the stability or instability of discrete dark solitons from the limiting configuration and confirm this criterion numerically. We also develop asymptotic calculations of the relevant eigenvalues for a number of prototypical configurations and illustrate their good agreement with the numerical data.