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Those massive stars that, during their deaths, give rise to gamma-ray bursts (GRBs) must be endowed with an unusually large amount of angular momentum in their inner regions, one to two orders of magnitude greater than the ones that make common pulsars. Yet the inclusion of mass loss and angular momentum transport by magnetic torques during the precollapse evolution is known to sap the core of the necessary rotation. Here we explore the evolution of very rapidly rotating, massive stars, including stripped down helium cores that might result from mergers or mass transfer in a binary, and single stars that rotate unusually rapidly on the main sequence. For the highest possible rotation rates (about 400 km/s), a novel sort of evolution is encountered in which single stars mix completely on the main sequence, never becoming red giants. Such stars, essentially massive "blue stragglers", produce helium-oxygen cores that rotate unusually rapidly. Such stars might comprise roughly 1% of all stars above 10 solar masses and can, under certain circumstances retain enough angular momentum to make GRBs. Because this possibility is very sensitive to mass loss, GRBs will be much more probable in regions of low metallicity.