In 1982 Feynman(1) observed that quantum-mechanical systems have an information-processing capability much greater than that of corresponding classical systems, and could thus potentially be used to implement a new type of powerful computer. Three years later Deutsch(2) described a quantum-mechanical Turing machine, showing that quantum computers could indeed be constructed. Since then there has been extensive research in this field, but although the theory is fairly well understood, actually building a quantum computer has proved extremely difficult. Only two methods have been used to demonstrate quantum logic gates : ion traps(3,4) and nuclear magnetic resonance (NMR)(5,6). NMR quantum computers have recently been used to solve a simplequantum algorithm-the two-bit Deutsch problem(7,8). Here we show experimentally that such a computer can be used to implement a non-trivial fast quantum search algorithm initially developed by Grover(9,10), which can be conducted faster than a comparable search on a classical computer.