  A binned Dalitz plot analysis of $B^\pm \to D K^\pm$ decays, with $D \to
K^0_{\rm S} \pi^+\pi^-$ and $D \to K^0_{\rm S} K^+ K^-$, is performed to
measure the CP-violating observables $x_{\pm}$ and $y_{\pm}$ which are
sensitive to the CKM angle $\gamma$. The analysis exploits 1.0 $\rm fb^{-1}$ of
data collected by the LHCb experiment. The study makes no model-based
assumption on the variation of the strong phase of the D decay amplitude over
the Dalitz plot, but uses measurements of this quantity from CLEO-c as input.
The values of the parameters are found to be $x_- = (0.0 \pm 4.3 \pm 1.5 \pm
0.6) \times 10^{-2}$, $y_- = (2.7 \pm 5.2 \pm 0.8 \pm 2.3) \times 10^{-2}$,
$x_+ = (-10.3 \pm 4.5 \pm 1.8 \pm 1.4)\times 10^{-2}$ and $y_+ = (-0.9 \pm 3.7
\pm 0.8 \pm 3.0)\times 10^{-2}$. The first, second, and third uncertainties are
the statistical, the experimental systematic, and the error associated with the
precision of the strong-phase parameters measured at CLEO-c, respectively.
These results correspond to $\gamma = (44^{\,+43}_{\,-38})^\circ$, with a
second solution at $\gamma \to \gamma + 180^\circ$, and $r_B = 0.07 \pm 0.04$,
where $r_B$ is the ratio between the suppressed and favoured B decay
amplitudes.
