We report on a terahertz spectrometer for high-resolution molecular spectroscopy based on a quantum-cascade laser. High-frequency modulation (up to 50 MHz) of the laser driving current produces a simultaneous modulation of the frequency and amplitude of the laser output. The modulation generates sidebands, which are symmetrically positioned with respect to the laser carrier frequency. The molecular transition is probed by scanning the sidebands across it. In this way, the absorption and the dispersion caused by the molecular transition are measured. The signals are modeled by taking into account the simultaneous modulation of the frequency and amplitude of the laser emission. This allows for the determination of the strength of the frequency as well as amplitude modulation of the laser and of molecular parameters such as pressure broadening.