  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 (UHE) cosmic rays shows a sharp suppression at an energy of $6
\times 10^{19}$ eV, consistent with the expected cutoff energy. We observe the
``ankle'' of the cosmic-ray energy spectrum as well, at an energy of $4 \times
10^{18}$ eV. We describe the experiment, data collection, analysis, and
estimate the systematic uncertainties. The results are presented and the
calculation of the statistical significance of our observation is described.
