Weak motif discovery is a fundamental problem in computational biology. It is difficult to solve because the number of mismatches between the true motif and its mutated instances is so large that the spurious signals may disguise the true ones. While many algorithms have been proposed to solve this problem, they either require a large amount of memory or consume too much time. In this paper, a sample-driven algorithm, ListMotif, is proposed, which uses the substrings from the data to construct lists of motif instances. ListMotif is memory efficient and by avoiding re-calculations on the hamming distances between the substrings it also exhibits time efficiency. The experiments on real biological data have demonstrated its applicability in practice. Meanwhile, the test results on synthetic data show that ListMotif is able to discover longer and weaker motifs compared to some previously proposed algorithms.