A sample of 65210 K ± → π 0 π 0 e ± ν (K e4 00 ) decay candidates with 1% background contamination has been collected in 2003-2004 by the NA48/2 collaboration at the CERN SPS. A study of the differential rate provides the first measurement of the hadronic form factor variation in the plane (M ππ 2 , M eν 2 ) and brings evidence for a cusp-like structure in the distribution of the squared π 0 π 0 invariant mass around 4m2π+ . Exploiting a model independent description of this form factor, the branching ratio, inclusive of radiative decays, is obtained using the K ± → π 0 π 0 π ± decay mode as normalization. It is measured to be BR(K e4 00 ) = (2.552 ± 0.010stat ± 0.010syst ± 0.032ext) × 10−5, which improves the current world average precision by an order of magnitude while the 1.4% relative precision is dominated by the external uncertainty from the normalization mode. A comparison with the properties of the corresponding mode involving a π + π − pair (K e4 + − ) is also presented.