Depth of maximum of air-shower profiles at the Pierre Auger Observatory. I. Measurements at energies above<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>17.8</mml:mn></mml:mrow></mml:msup><mml:mtext>  </mml:mtext><mml:mi>eV</mml:mi></mml:mrow></mml:math>

We report a study of the distributions of the depth of maximum, ${X}_{\mathrm{max}}$, of extensive air-shower profiles with energies above $1{0}^{17.8}\text{ }\text{ }\mathrm{eV}$ as observed with the fluorescence telescopes of the Pierre Auger Observatory. The analysis method for selecting a data sample with minimal sampling bias is described in detail as well as the experimental cross-checks and systematic uncertainties. Furthermore, we discuss the detector acceptance and the resolution of the ${X}_{\mathrm{max}}$ measurement and provide parametrizations thereof as a function of energy. The energy dependence of the mean and standard deviation of the ${X}_{\mathrm{max}}$ distributions are compared to air-shower simulations for different nuclear primaries and interpreted in terms of the mean and variance of the logarithmic mass distribution at the top of the atmosphere.