We present measurements of the branching fractions for the charmless two-body decays B0→π+π- and B0→K+π-, and a search for the decay B0→K+K-. We include the effects of final-state radiation from the daughter mesons for the first time, and quote branching fractions for the inclusive processes B0→h+h′-nγ, where h and h′ are pions or kaons. The maximum value of the sum of the energies of the n undetected photons, Eγmax, is mode-dependent. Using a data sample of approximately 227×106 Υ(4S)→BB̄ decays collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at SLAC, we measure: B(B0→π+π-nγ;Eγmax =150MeV)=(5.1±0.4±0.2) ×10-6, B(B0→K+π-nγ;Eγmax =105MeV)=(18.1±0. 6±0.6)×10-6, B(B0→K+K-nγ;Eγmax =59MeV)<0.5×10-6(90%confidence level), where the first uncertainty is statistical and the second is systematic. Theoretical calculations can be used to extrapolate from the above measurements the nonradiative branching fractions, B0. Using one such calculation, we find: B0(B0→π+π-)=(5.5±0. 4±0.3)×10-6, B0(B0→K+π-)=(19.1±0.6±0.6) ×10-6, B0(B0→K+K-)<0.5×10-6(90%confidence level). Meaningful comparison between theory and experiment, as well as combination of measurements from different experiments, can be performed only in terms of these nonradiative quantities. © 2007 The American Physical Society.