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We present a measurement of the time-dependent $CP$ asymmetry for the neutral $B$-meson decay into the $CP=+1$ final state ${K}_{S}^{0}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$, with ${K}_{S}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. We use a sample of approximately 227 million $B$-meson pairs recorded at the $\mathit{\ensuremath{\Upsilon}}(4S)$ resonance with the BABAR detector at the PEP-II $B$-Factory at SLAC. From an unbinned …