Proton-proton collision data recorded in 2011 and 2012 by the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb(-1), are analysed to search for the charmless B-0 -> rho(0)rho(0) decay. More than 600 B-0 -> (pi(+)pi(-))(pi(+)pi(-)) signal decays are selected and used to perform an amplitude, analysis, under the assumption of no CP violation in the decay, from which the B-0 -> rho(0)rho(0) decay is observed for the first time with 7.1 standard deviations significance. The fraction of B-0 -> rho(0)rho(0) decays yielding a longitudinally polarised final state is measured to be f(L) = 0.745(-0.058)(+0.048)(stat) +/- 0.034(syst). The B-0 -> rho(0)rho(0) branching fraction, using the B-0 -> phi K*(892)(0) decay as reference, is also reported as B(B-0 -> rho(0)rho(0)) = (0.94 +/- 0.17(stat) +/- 0.09(syst) +/- 0.06(BF)) x 10(-6). (C) 2015 CERN for the benefit of the LHCb Collaboration. Published by Elsevier B.V. This is an open access article under the CC BY license