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We describe the measurement of the depth of maximum, Xmax, of the longitudinal development of air showers induced by cosmic rays. Almost 4000 events above 1018 eV observed by the fluorescence detector of the Pierre Auger Observatory in coincidence with at least one surface detector station are selected for the analysis. The average shower maximum was found to evolve with energy at a rate of (106+35-21) g/cm2/decade below 1018.24 ±0.05 eV, and (24 ± 3) g/cm2/decade above this energy. The measured shower-to-shower fluctuations decrease from about 55 to 26 g/cm2. The interpretation of these results in terms of the cosmic ray mass composition is briefly discussed.