We report the measurement of the Cabibbo-Kobayashi-Maskawa CP-violating angle $\gamma$ through a Dalitz plot analysis of neutral $D$ meson decays to $K_S^0 \pi^+ \pi^-$ and $K_S^0 K^+ K^-$ produced in the processes $B^\mp \to D K^\mp$, $B^\mp \to D^{*} K^\mp$ with $D^* \to D \pi^0,D \gamma$, and $B^\mp \to D K^{*\mp}$ with $K^{*\mp} \to K_S^0 \pi^\mp$, using 468 million $B\bar{B}$ pairs collected by the BABAR detector at the PEP-II asymmetric-energy $e^+e^-$ collider at SLAC. We measure $\gamma=(68 \pm 14 \pm 4 \pm 3)^\circ$ ({\rm mod $180^\circ$}), where the first error is statistical, the second is the experimental systematic uncertainty and the third reflects the uncertainty in the description of the neutral $D$ decay amplitudes. This result is inconsistent with $\gamma = 0$ (no direct CP violation) with a significance of 3.5 standard deviations. Show Partial Abstract