  We study the decay B0 --> rho0 rho0 in a sample of about 427 million
Upsilon(4S) --> BBbar decays collected with the BABAR detector at the PEP-II
asymmetric-energy e+e- collider at SLAC. We find the branching fraction
  B = (0.84 +/- 0.29 +/- 0.17)*1e-6 and longitudinal polarization fraction of
f_L = 0.70 +/- 0.14 +/- 0.05, where the first uncertainty is statistical, and
the second is systematic. The evidence for the B0 --> rho0 rho0 signal has 3.6
sigma significance. We investigate the proper-time dependence of the
longitudinal component in the decay and measure the CP-violating coefficients
S^{00}_L = 0.5 +/- 0.9 +/- 0.2 and C^{00}_L = 0.4 +/- 0.9 +/- 0.2,
corresponding to the sine and cosine terms in the time evolution of asymmetry.
We study the implication of these results for penguin contributions in B -->
rho rho decays and for the CKM unitarity angle alpha.
