We present a study of the decay B0-->pi(0)pi(0) based on a sample of 124 x 10(6) BB pairs recorded by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. We observe 46+/-13+/-3 events, where the first error is statistical and the second is systematic, corresponding to a significance of 4.2 standard deviations including systematic uncertainties. We measure the branching fraction B(B0-->pi(0)pi(0))=(2.1+/-0.6+/-0.3)x10(-6), averaged over B0 and B(0) decays.