Laser cycling of resonances can remove entropy from a system via spontaneously emitted photons, with electronic resonances providing the fastest cooling timescales because of their rapid spontaneous relaxation. Although atoms are routinely laser-cooled, even simple molecules pose two interrelated challenges for cooling: every populated rotational-vibrational state requires a different laser frequency, and electronic relaxation generally excites vibrations. Here we cool trapped AlH(+) molecules to their ground rotational-vibrational quantum state using an electronically exciting broadband laser to simultaneously drive cooling resonances from many different rotational levels. Undesired vibrational excitation is avoided because of vibrational-electronic decoupling in AlH(+). We demonstrate rotational cooling on the 140(20) ms timescale from room temperature to 3.8(-0.3)(+0.9) K, with the ground-state population increasing from ~3 to 95.4(-2.1)(+1.3)%. This cooling technique could be applied to several other neutral and charged molecular species useful for quantum information processing, ultracold chemistry applications and precision tests of fundamental symmetries.