A method to achieve high-brightness self-amplified spontaneous emission (HB-SASE) in the free-electron laser (FEL) is described. The method uses repeated nonequal electron beam delays to delocalize the collective FEL interaction and break the radiation coherence length dependence on the FEL cooperation length. The method requires no external seeding or photon optics and so is applicable at any wavelength or repetition rate. It is demonstrated, using linear theory and numerical simulations, that the radiation coherence length can be increased by approximately 2 orders of magnitude over SASE with a corresponding increase in spectral brightness. Examples are shown of HB-SASE generating transform-limited FEL pulses in the soft x-ray and near transform-limited pulses in the hard x-ray. Such pulses may greatly benefit existing applications and may also open up new areas of scientific research.