We demonstrate that the electronic gap of a graphene bilayer can be controlled externally by applying a gate bias. From the magnetotransport data (Shubnikov-de Haas measurements of the cyclotron mass), and using a tight-binding model, we extract the value of the gap as a function of the electronic density. We show that the gap can be changed from zero to midinfrared energies by using fields of less, approximately < 1 V/nm, below the electric breakdown of SiO2. The opening of a gap is clearly seen in the quantum Hall regime.