A measurement of the cosmic-ray spectrum for energies exceeding 4×1018 eV is presented, which is based on the analysis of showers with zenith angles greater than 60° detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above 5.3×1018 eV, the ``ankle'', the flux can be described by a power law E−γ with index γ=2.70 ± 0.02 (stat) ± 0.1 (sys) followed by a smooth suppression region. For the energy (Es) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find Es=(5.12±0.25 (stat)+1.0−1.2 (sys))×1019 eV.