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We present a combined theoretical and experimental study of the ferromagnetic semiconductor (Ga,Mn)As which explains the remarkably large changes observed on low-temperature annealing. Careful control of the annealing conditions allows us to obtain samples with ferromagnetic transition temperatures up to 159 K. Ab initio calculations, in situ Auger spectroscopy, and resistivity measurements during annealing show that the observed changes are due to out diffusion of Mn interstitials towards the surface, governed by an energy barrier of 0.7-0.8 eV. Electric fields induced by Mn acceptors have a significant effect on the diffusion.