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We study the existence and stability of the standing waves for the periodiccubic nonlinear Schr\"odinger equation with a point defect determined by aperiodic Dirac distribution at the origin. This equation admits a smooth curveof positive periodic solutions in the form of standing waves with a profilegiven by the Jacobi elliptic function of dnoidal type. Via a perturbationmethod and continuation argument, we obtain that in the case of an attractivedefect the standing wave solutions are stable in $H^1_{per}$ with respect toperturbations which have the same period as the wave itself. In the case of arepulsive defect, the standing wave solutions are stable in the subspace ofeven functions of $H^1_{per}$ and unstable in $H^1_{per}$ with respect toperturbations which have the same period as the wave itself.

The Non-Linear Schr\"odinger Equation with a periodic {\bf{$\delta$}}--interaction