  We report experimental details and results of a new measurement of the decay
K+ -> pi+pi-e+nue (Ke4}. A sample of more than 400,000 Ke4 events with low
background has been collected by Experiment 865 at the Brookhaven Alternate
Gradient Synchrotron. From these data, the branching ratio
(4.11+-0.01+-0.11)*10**(-5) and the pi-pi invariant mass dependence of the form
factors F, G, and H of the weak hadronic current as well as the phase shift
difference delta00-delta11 for pi-pi-scattering were extracted. Using
constraints based on analyticity and chiral symmetry, a new value with
considerably improved accuracy for the s-wave pi-pi-scattering length a00 has
been obtained also: a00=0.216+-0.013 (stat.)+-0.002(syst.)+-0.002(theor.).
