Nuclear-magnetic-resonance ~NMR! quantum computation is a fruitful arena in which to develop and demonstrate an enhanced capability for quantum control over molecular systems, regardless of the prospects, which may be limited, for building a quantum computer superior to a conventional computer for any computing task. We demonstrate a five-bit NMR quantum computer that distinguishes among various functions on four bits, making use of quantum parallelism, an example of the Deutsch-Jozsa problem. Its construction draws on the recognition of the sufficiency of linear coupling along a chain of nuclear spins, the synthesis of a suitably coupled molecule containing four distinct nuclear species, and the use of a multichannel spectrometer. Radiofrequency pulse sequences are described to execute controlled-NOT gates on two adjoining spins while leaving the other three spins essentially unaffected.