Graphene is an attractive material for use in optical detectors because it absorbs light from mid-infrared to ultraviolet wavelengths with nearly equal strength. Graphene is particularly well suited for bolometers-devices that detect temperature-induced changes in electrical conductivity caused by the absorption of light-because its small electron heat capacity and weak electron-phonon coupling lead to large light-induced changes in electron temperature. Here, we demonstrate a hot-electron bolometer made of bilayer graphene that is dual-gated to create a tunable bandgap and electron-temperature-dependent conductivity. The bolometer exhibits a noise-equivalent power (33 fW Hz(-1/2) at 5 K) that is several times lower, and intrinsic speed (>1 GHz at 10 K) three to five orders of magnitude higher than commercial silicon bolometers and superconducting transition-edge sensors at similar temperatures.