  We study B0->rho0rho0 decays in a sample of 465*10^6 Upsilon (4S)->BBbar
events collected with the BABAR detector at the PEP-II asymmetric-energy e+e-
collider located at the Stanford Linear Accelerator Center (SLAC). We measure
the branching fraction B = (0.92 +/- 0.32 +/- 0.14)*10^{-6} and longitudinal
polarization fraction f_L = 0.75^{+0.11}_{-0.14} +/- 0.04, where the first
uncertainty is statistical and the second is systematic. The evidence for the
B0->rho0rho0 signal has a significance of 3.1 standard deviations, including
systematic uncertainties. We investigate the proper-time dependence of the
longitudinal component in the decay and measure the CP-violating coefficients
S^{00}_L = (0.3 +/- 0.7 +/- 0.2) and C^{00}_L = (0.2 +/- 0.8 +/- 0.3). We study
the implication of these results for the unitarity triangle angle alpha.
