The structure dependent radiative K+→e+νγ (Ke2γSD+) decay was investigated with stopped positive kaons. The e+ momentum spectra containing 574±30Ke2γSD+ events with a K+→μ+ν (Kμ2) background of 28±19 events were measured with and without a photon in coincidence and analyzed with Monte Carlo simulations for acceptance and detector response to extract the ratio of the branching ratio of the Ke2γSD+ decay and the K+→e+ν decay including the internal bremsstrahlung process (Ke2(γ)). A value of Br(Ke2γSD+)/Br(Ke2(γ))=1.12±0.07stat±0.04syst was obtained. This indicates a partial branching ratio, Br(Ke2γSD+,pe>200MeV/c,Eγ>10MeV)/Br(Kμ2)=(1.85±0.11stat±0.07syst)×10−5, which is 25% (∼2.5σ) higher than the previous experimental result.