  A large low-background sample of events (10300) has been collected for the
rare decay of kaons in flight K+ -> pi+ e+ e- by experiment E865 at the
Brookhaven AGS. The decay products were accepted by a broad band
high-resolution charged particle spectrometer with particle identification. The
branching ratio (2.94 +- 0.05(stat.) +- 0.13(syst.) +- 0.05(model))*10**{-7}
was determined normalizing to events from the decay chain K+ -> pi+ pi0; pi0 ->
e+ e- gamma. From the analysis of the decay distributions the vector nature of
this decay is firmly established now, and limits on scalar and tensor
contributions are deduced. From the (e+ e-) invariant mass distribution the
decay form factor f(z)=f0(1+ delta*z) (z=M(ee)**2/m(K)**2) is determined with
delta=2.14 +- 0.13 +- 0.15. Chiral QCD perturbation theory predictions for the
form factor are also tested, and terms beyond leading order O(p**4) are found
to be important.
