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In this letter, we propose the bilayer graphene tunnel field-effect transistor (TFET) as a device suitable for fabrication and circuit integration with present-day technology. It provides high I on/I off ratio at ultralow supply voltage, without the limitations in terms of prohibitive lithography and patterning requirements for circuit integration of graphene nanoribbons. Our investigation is based on the solution of the coupled Poisson and Schrodinger equations in three dimensions, within the non-equilibrium Green's function formalism on a tight binding Hamiltonian. We show that the small achievable gap of only few hundreds of millielectronvolts is still enough for promising TFET operation, providing a large I on/I off ratio in excess of 103 even for a supply voltage of only 0.1 V. A key to this performance is the low quantum capacitance of bilayer graphene, which permits to obtain an extremely small subthreshold swing S smaller than 20 mV/dec at room temperature.