The High Resolution Fly's Eye (HiRes) experiment has observed the Greisen-Zatsepin-Kuzmin suppression (called the GZK cutoff) with a statistical significance of five standard deviations. HiRes' measurement of the flux of ultrahigh energy cosmic rays shows a sharp suppression at an energy of 6 x 10(19) eV, consistent with the expected cutoff energy. We observe the ankle of the cosmic-ray energy spectrum as well, at an energy of 4 x 10(18) eV. We describe the experiment, data collection, and analysis and estimate the systematic uncertainties. The results are presented and the calculation of the statistical significance of our observation is described.