The authors report a final assessment of the energy spectrum above 4*10¹⁷ eV which has been produced from measurements made at the Haverah Park extensive air shower array between 1968 and 1987. The methods of analysis are described in some detail and they confirm that the slope of the energy spectrum from 4*10¹⁷ eV to 4*10¹⁸ eV is steeper than the value measured at lower energies, and that above 10¹⁹ eV the spectral slope flattens again. The integral intensity at 10¹⁸ eV is (1.8+or-0.1)*10^-12 m^-2 s^-1 sr^-1, or about 60 events km^-2 sr^-1 year^-1. They have demonstrated that there is no evidence for any cut-off in the spectrum at least to 10²² eV, close to the highest energy observed. At 10²⁰ eV their estimate of the integral intensity (based on only eight events) is (3+or-2)*10^-16 m^-2 s^-1 sr^-1, or about one event per km² per century per steradian. Thus to explore this energy range and the decade beyond in greater detail requires exposure rates of at least 1000 km² per year. They have compared their measurements with data from Akeno, Fly's Eye and Yakutsk.