  We report a measurement of time-dependent $CP$ violation in $B^0 \to K^0_S
\pi^0 \pi^0$ decays using a data sample of $772 \times 10^6$ $B\bar{B}$ pairs
collected by the Belle experiment runnin g at the $\Upsilon (4S)$ resonance at
the KEKB $e^+ e^-$ collider. This decay proceeds mainly via a $b\to sd\bar{d}$
"penguin" amplitude. The results are $\sin 2\phi^{\rm eff}_1 =
0.92^{+0.27}_{-0.31}~$ (stat.) $\pm 0.11$ (syst.) and $\mathcal{A} = 0.28 \pm
0.21$ (stat.) $\pm 0.04$ (syst.), which are the most precise measurements of
$CP$ violati on in this decay mode to date. The value for the $CP$-violating
parameter $\sin 2\phi^{\rm eff}_1$ is consistent with that obtained using decay
modes proceeding via a $b\to c\bar{c}s$ "tree" amplitude.
