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The first observation of the B-s(0) -> eta'eta' decay is reported. The study is based on a sample of proton-proton collisions corresponding to 3.0 fb(-1) of integrated luminosity collected with the LHCb detector. The significance of the signal is 6.4 standard deviations. The branching fraction is measured to be [3.31 +/- 0.64(stat) +/- 0.28(syst) +/- 0.12(norm)] x 10(-5), where the third uncertainty comes from the B-+/- -> eta'K-+/- branching fraction that is used as a normalization. In addition, the charge asymmetries of B-+/- -> eta'K-+/- and B-+/- -> phi K-+/-, which are control channels, are measured to be (-0.2 +/- 1.3)% and (+1.7 +/- 1.3)%, respectively. All results are consistent with theoretical expectations.