The spectrum of amplification of terahertz radiation in a structured active graphene bilayer consisting of two identical periodic graphene microribbon arrays separated by a thin dielectric barrier layer is theoretically investigated. The system supports optical and acoustic plasmon modes. The resonant frequencies of the optical and acoustic modes change oppositely with the dielectric-layer thickness, which allows plasmon-mode anticrossing. It is shown that the investigated graphene structure is characterized by a strong plasmon response and giant terahertz-radiation amplification at plasma resonance frequencies in the vicinity of the anticrossing between the optical and acoustic plasmon modes at room temperature.