We have developed a frequency-modulated, tunable, amplitude-squeezed, diode-laser-based source and used it to perform FM spectroscopy on rubidium. The setup consists of a free-running diode laser injection locked by a frequency-stabilized, current-modulated diode laser. The injection-locked slave laser beam adopted the frequency spectrum of the master laser beam while rejecting residual AM in the master laser beam by more than 50 dB. Injection locking also enhanced amplitude squeezing in the slave laser beam by suppressing uncorrelated longitudinal sidemodes. The noise floor of the measurement was 0.8 dB below the shot-noise level.