The potential of graphene-based materials consisting of one or a few layers of graphite for integrated electronics originates from the large room-temperature carrier mobility in these systems (approximately 10,000 cm2 V(-1) s(-1)). However, the realization of electronic devices such as field-effect transistors will require controlling and even switching off the electrical conductivity by means of gate electrodes, which is made difficult by the absence of a bandgap in the intrinsic material. Here, we demonstrate the controlled induction of an insulating state--with large suppression of the conductivity--in bilayer graphene, by using a double-gate device configuration that enables an electric field to be applied perpendicular to the plane. The dependence of the resistance on temperature and electric field, and the absence of any effect in a single-layer device, strongly suggest that the gate-induced insulating state originates from the recently predicted opening of a bandgap between valence and conduction bands.