  The electronic structure of heavily overdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ is investigated by angle-resolved
photoemission spectroscopy. The long-sought bilayer band splitting in this
two-plane system is observed in both normal and superconducting states, which
qualitatively agrees with the bilayer Hubbard model calculations. The maximum
bilayer energy splitting is about 88 meV for the normal state feature, while it
is only about 20 meV for the superconducting peak. This anomalous behavior
cannot be reconciled with the quasiparticle picture.
