An experimental search for an electric dipole moment (EDM) of the neutron has been carried out at the Institut Laue-Langevin, Grenoble. Spurious signals from magnetic-field fluctuations were reduced to insignificance by the use of a cohabiting atomic-mercury magnetometer. Systematic uncertainties, including geometric-phase-induced false EDMs, have been carefully studied. The results may be interpreted as an upper limit on the neutron EDM of |d(n)|< 2.9 x 10(-26)e cm (90% C.L.).