Evidence for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>CP</mml:mi></mml:math>-violating asymmetries in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>→</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mi>−</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>decays and constraints on the CKM angle<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

We present an improved measurement of $\mathrm{CP}$-violating asymmetries in ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays based on a $78{\mathrm{fb}}^{\ensuremath{-}1}$ data sample collected at the $\ensuremath{\Upsilon}(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider. We reconstruct one neutral B meson as a ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ $\mathrm{CP}$ eigenstate and identify the flavor of the accompanying B meson from inclusive properties of its decay products. We apply an unbinned maximum likelihood fit to the distribution of the time intervals between the two B meson decay points. The fit yields the $\mathrm{CP}$-violating asymmetry amplitudes ${\mathcal{A}}_{\ensuremath{\pi}\ensuremath{\pi}}=+0.77\ifmmode\pm\else\textpm\fi{}0.27(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.08(\mathrm{syst})$ and ${\mathcal{S}}_{\ensuremath{\pi}\ensuremath{\pi}}=\ensuremath{-}1.23\ifmmode\pm\else\textpm\fi{}0.41(\mathrm{stat}){}_{\ensuremath{-}0.07}^{+0.08}(\mathrm{syst}),$ where the statistical uncertainties are determined from the Monte Carlo pseudoexperiments. We obtain confidence intervals for $\mathrm{CP}$-violating asymmetry parameters ${\mathcal{A}}_{\ensuremath{\pi}\ensuremath{\pi}}$ and ${\mathcal{S}}_{\ensuremath{\pi}\ensuremath{\pi}}$ based on a frequentist approach. We rule out the $\mathrm{CP}$-conserving case, ${\mathcal{A}}_{\ensuremath{\pi}\ensuremath{\pi}}={\mathcal{S}}_{\ensuremath{\pi}\ensuremath{\pi}}=0,$ at the $99.93%$ confidence level. We discuss how these results constrain the value of the Cabibbo-Kobayashi-Maskawa (CKM) angle ${\ensuremath{\varphi}}_{2}.$