  Proton-proton collision data recorded in 2011 and 2012 by the \lhcb
experiment, co\-rres\-pon\-ding to an integrated luminosity of 3.0\invfb, 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 ${\BF(B^0 \to \rho^0 \rho^0) = (0.94 \pm 0.17 ({\rm stat}) \pm 0.09
({\rm syst}) \pm 0.06 ({\rm BF})) \times 10^{-6}}$.
