  The rare decay K0(L) --> pi0 pi0 nu nu-bar was studied with the E391a
detector at the KEK 12-GeV proton synchrotron. Based on 9.4 x 10^9 K0L decays,
an upper limit of 8.1 x 10^{-7} was obtained for the branching fraction at 90%
confidence level. We also set a limit on the K0(L) --> pi0 pi0 X (X -->
invisible particles) process; the limit on the branching fraction varied from
7.0 x 10^{-7} to 4.0 x 10^{-5} for the mass of X ranging from 50 MeV/c^2 to 200
MeV/c^2.
