A search for the B0→K+K− decay is performed using pp-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 B0s→π+π− decay. The measured branching fractions are B(B0→K+K−)=(7.80±1.27±0.81±0.21)×10−8 and B(B0s→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 B0s mode, the fourth accounts for the uncertainty on the ratio of the probabilities for b quarks to hadronize into B0s and B0 mesons.