We present a compelling evidence for the opening of a bandgap in exfoliated bottom-gated bilayer graphene by fitting the gate-voltage modulated infrared reflectivity spectra in a large range of doping levels with a tight-binding model and the Kubo formula. A close quantitative agreement between the experimental and calculated spectra is achieved, allowing us to determine self-consistently the full set of Slonczewski-Weiss-McClure tight-binding parameters together with the gate-voltage dependent

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... bandgap. The doping dependence of the bandgap shows a good agreement with the existing calculations that take the effects of self-screening into account. We also identify certain mismatches between the tight-binding model and the data, which can be related to electron-electron and electron-phonon interactions.