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This letter demonstrates a 1-GHz hybrid mode-locked monolithic semiconductor laser fabricated on indium phosphide. Its operating regimes are explored and optical pulses as short as 36 ps were measured. The linear cavity is 41 mm long with integrated active quantum well and passive waveguide structures. To our knowledge, this is the lowest reported repetition rate for a monolithically integrated mode-locked semiconductor laser. We further describe optimization steps of the saturable absorber reverse bias, driving RF frequency, and the semiconductor optical amplifier gain current for minimal output pulsewidth.