The electronic bandgap is an intrinsic property of semiconductors and insulators that largely determines their transport and optical properties. As such, it has a central role in modern device physics and technology and governs the operation of semiconductor devices such as p-n junctions, transistors, photodiodes and lasers. A tunable bandgap would be highly desirable because it would allow great flexibility in design and optimization of such devices, in particular if it could be tuned by applying a variable external electric field. However, in conventional materials, the bandgap is fixed by their crystalline structure, preventing such bandgap control. Here we demonstrate the realization of a widely tunable electronic bandgap in electrically gated bilayer graphene. Using a dual-gate bilayer graphene field-effect transistor (FET) and infrared microspectroscopy, we demonstrate a gate-controlled, continuously tunable bandgap of up to 250 meV. Our technique avoids uncontrolled chemical doping and provides direct evidence of a widely tunable bandgap-spanning a spectral range from zero to mid-infrared-that has eluded previous attempts. Combined with the remarkable electrical transport properties of such systems, this electrostatic bandgap control suggests novel nanoelectronic and nanophotonic device applications based on graphene.