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.