Single-crystal superconductors of the general formula (LRE)-Ba-Cu-O (light rare earth, LRE = Nd, Sm, Eu and Gd) have considerable potential for engineering applications because of their ability to trap magnetic fields significantly higher than those achievable with permanent magnets. But the lack of a process by which these materials can be fabricated reliably and economically in the form of large single grains has severely hindered their development. We report a practical processing method for the fabrication in air of single-crystal (RE)BCO. The technique is economical and offers considerable freedom in terms of the processing parameters and reproducibility in growth of oriented single grains. The process is based primarily on the development of a new type of generic seed crystal that can effectively promote the epitaxial nucleation of any (RE)BCO system, and secondly on suppressing the formation of RE-Ba solid solution in a controlled manner within large grains processed in air.