A search for the B0→K+K- decay is performed using p p -collision data collected by LHCb. The data set corresponds to integrated luminosities of 1.0 and 2.0 fb-1 at center-of-mass energies of 7 and 8 TeV, respectively. This decay is observed for the first time, with a significance of more than 5 standard deviations. The analysis also results in an improved measurement of the branching fraction for the Bs0→π+π- decay. The measured branching fractions are B (B0→K+K- )=(7.80 ±1.27 ±0.81 ±0.21 )×10-8 and B (Bs0→π+π- )=(6.91 ±0.54 ±0.63 ±0.19 ±0.40 )×10-7 . The first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the B0→K+π- branching fraction used as a normalization. For the Bs0 mode, the fourth accounts for the uncertainty on the ratio of the probabilities for b quarks to hadronize into Bs0 and B0 mesons.