High statistics measurement of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mn>4</mml:mn><mml:mn/></mml:mrow></mml:msub></mml:mrow></mml:math>decay properties

We report experimental details and results of a new measurement of the decay ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}{(K}_{e4}).$ A sample of more than $400,000{K}_{e4}$ events with low background has been collected by Experiment 865 at the Brookhaven Alternate Gradient Synchrotron. From these data, the branching ratio $(4.11\ifmmode\pm\else\textpm\fi{}0.01\ifmmode\pm\else\textpm\fi{}0.11)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and the $\ensuremath{\pi}\ensuremath{\pi}$ invariant mass dependence of the form factors $F,G,$ and H of the weak hadronic current as well as the phase shift difference ${\ensuremath{\delta}}_{0}^{0}\ensuremath{-}{\ensuremath{\delta}}_{1}^{1}$ for $\ensuremath{\pi}\ensuremath{\pi}$ scattering were extracted. Using constraints based on analyticity and chiral symmetry, a new value with considerably improved accuracy for the s-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering length ${a}_{0}^{0}$ has been obtained also: ${a}_{0}^{0}=0.216\ifmmode\pm\else\textpm\fi{}0.013(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.002(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.002(\mathrm{theor}).$