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Measurement of the branching fraction and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>-violating asymmetry for the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>

Measurement of the branching fraction and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>-violating asymmetry for the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>

We measure the branching fraction and the $CP$-violating asymmetry of ${B}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{0}$ decays with $227$ million $\ensuremath{\Upsilon}(4S)\ensuremath{\rightarrow}B\overline{B}$ events collected with the BABAR detector at the PEP-II asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider at SLAC. We obtain a branching fraction $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\pi}}^{0})=(11.4\ifmmode\pm\else\textpm\fi{}0.9\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $CP$-violating asymmetry parameters ${C}_{{K}_{S}^{0}}{\ensuremath{\pi}}^{0}=0.06\ifmmode\pm\else\textpm\fi{}0.18\ifmmode\pm\else\textpm\fi{}0.03$ and ${S}_{{K}_{S}^{0}}{\ensuremath{\pi}}^{0}={0.35}_{\ensuremath{-}0.33}^{+0.30}\ifmmode\pm\else\textpm\fi{}0.04$, where the first error is statistical and …