We report the results of a high-spatial-resolution search for the progenitor of type Ic supernova SN 2004gt, using the newly commissioned Keck laser-guide star adaptive optics system (LGSAO) along with archival Hubble Space Telescope data. This is the deepest search yet performed for the progenitor of any type Ib/c event in a wide wavelength range stretching from the far UV to the near IR. We determine that the progenitor of SN 2004gt was most likely less luminous than M V = −5.5 and M B = −6.5 magnitudes. The massive stars exploding as hydrogen-deficient core-collapse supernovae (SNe) should have lost their outer hydrogen envelopes prior to their explosion, either through winds – such stars are identified within our Galaxy as as Wolf-Rayet (W-R) stars – or to a binary companion. The luminosity limits we set rule out more than half of the known galactic W-R stars as possible progenitors of this event. In particular, they imply that a W-R progenitor should have been among the more-evolved (highly stripped, less luminous) of these stars, a concrete constraint on its evolutionary state just prior to core collapse. The possibility of a less luminous, lower-mass binary progenitor cannot be constrained. This study demonstrates the power of LGS observations in furthering our understanding of core collapse, and the physics powering supernovae, GRBs and XRFs.