  We study the existence and stability of the standing waves for the periodic
cubic nonlinear Schr\"odinger equation with a point defect determined by a
periodic Dirac distribution at the origin. This equation admits a smooth curve
of positive periodic solutions in the form of standing waves with a profile
given by the Jacobi elliptic function of dnoidal type. Via a perturbation
method and continuation argument, we obtain that in the case of an attractive
defect the standing wave solutions are stable in $H^1_{per}$ with respect to
perturbations which have the same period as the wave itself. In the case of a
repulsive defect, the standing wave solutions are stable in the subspace of
even functions of $H^1_{per}$ and unstable in $H^1_{per}$ with respect to
perturbations which have the same period as the wave itself.
