The DNA motif discovery problem abstracts the task of discovering short, conserved sites in genomic DNA. Pevzner and Sze recently described a precise combinatorial formulation of motif discovery that motivates the following algorithmic challenge: find twenty planted occurrences of a motif of length fifteen in roughly twelve kilobases of genomic sequence, where each occurrence of the motif differs from its consensus in four randomly chosen positions. Such "subtle" motifs, though statistically highly significant, expose a weakness in existing motif-finding algorithms, which typically fail to discover them. Pevzner and Sze introduced new algorithms to solve their (15,4)-motif challenge, but these methods do not scale efficiently to more difficult problems in the same family, such as the (14,4)-, (16,5)-, and (18,6)-motif problems. We introduce a novel motif-discovery algorithm, PROJECTION, designed to enhance the performance of existing motif finders using random projections of the input's substrings. Experiments on synthetic data demonstrate that PROJECTION remedies the weakness observed in existing algorithms, typically solving the difficult (14,4)-, (16,5)-, and (18,6)-motif problems. Our algorithm is robust to nonuniform background sequence distributions and scales to larger amounts of sequence than that specified in the original challenge. A probabilistic estimate suggests that related motif-finding problems that PROJECTION fails to solve are in all likelihood inherently intractable. We also test the performance of our algorithm on realistic biological examples, including transcription factor binding sites in eukaryotes and ribosome binding sites in prokaryotes.