It has been predicted that application of a strong electric field perpendicular to the plane of bilayer graphene can induce a significant band gap. We have measured the optical conductivity of bilayer graphene with an efficient electrolyte top gate for a photon energy range of 0.2-0.7 eV. We see the emergence of new transitions as a band gap opens. A band gap approaching 200 meV is observed when an electric field approximately 1 V/nm is applied, inducing a carrier density of about 10(13) cm(-2)}. The magnitude of the band gap and the features observed in the infrared conductivity spectra are broadly compatible with calculations within a tight-binding model.