Proton-proton collision data recorded in 2011 and 2012 by the lhcb experiment, co-rres-pon-ding to an integrated luminosity of 3.0invfb, are a-na-lysed to search for the charmless ${B^0 to rho^0 rho^0}$ decay. More than 600 ${B^0 to (pi^+pi^-)(pi^+pi^-)}$ signal decays are selected and used to perform an amplitude analysis from which the ${B^0 to rho^0 rho^0}$ decay is observed for the first time with 7.1 standard deviations significance. The fraction of ${B^0 to rho^0 rho^0}$ decays yielding a longitudinally polarised final state is measured to be $fL = 0.745^{+0.048}_{-0.058} ({rm stat}) pm 0.034 ({rm syst})$. The ${B^0 to rho^0 rho^0}$ branching fraction, using the ${B^0 to phi K^*(892)^{0}}$ decay as reference, is also reported as ${BF(B^0 to rho^0 rho^0) = (0.94 pm 0.17 ({rm stat}) pm 0.09 ({rm syst}) pm 0.06 ({rm BF})) times 10^{-6}}$.
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