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We report a measurement of the flux of cosmic rays with unprecedented precision and statistics using the Pierre Auger Observatory. Based on fluorescence observations in coincidence with at least one surface detector we derive a spectrum for energies above 10^18 eV. We also update the previously published energy spectrum obtained with the surface detector array. The two spectra are combined addressing the systematic uncertainties and, in particular, the influence of the energy resolution on the spectral shape. The spectrum can be described by a broken power law E^-gamma with index gamma=3.3 below the ankle which is measured at log10(E/eV) = 18.6. Above the ankle the spectrum is described by a power law with index 2.6 followed by a flux suppression, above about log10(E/eV) = 19.5, detected with high statistical significance.