Context. Gaia Data Release 1 (DR1) contains astrometric results for more than 1 billion stars brighter than magnitude 20.7 based on observations collected by the Gaia satellite during the first 14 months of its operational phase. Aims. We give a brief overview of the astrometric content of the data release and of the model assumptions, data processing, and validation of the results. Methods. For stars in common with the Hipparcos and Tycho-2 catalogues, complete astrometric single-star solutions are obtained by incorporating positional information from the earlier catalogues. For other stars only their positions are obtained, essentially by neglecting their proper motions and parallaxes. The results are validated by an analysis of the residuals, through special validation runs, and by comparison with external data. Results. For about two million of the brighter stars (down to magnitude ∼11.5) we obtain positions, parallaxes, and proper motions to Hipparcos-type precision or better. For these stars, systematic errors depending for example on position and colour are at a level of ±0.3 milliarcsecond (mas). For the remaining stars we obtain positions at epoch J2015.0 accurate to ∼10 mas. Positions and proper motions are given in a reference frame that is aligned with the International Celestial Reference Frame (ICRF) to better than 0.1 mas at epoch J2015.0, and non-rotating with respect to ICRF to within 0.03 mas yr−1 . The Hipparcos reference frame is found to rotate with respect to the Gaia DR1 frame at a rate of 0.24 mas yr−1 . Conclusions. Based on less than a quarter of the nominal mission length and on very provisional and incomplete calibrations, the quality and completeness of the astrometric data in Gaia DR1 are far from what is expected for the final mission products. The present results nevertheless represent a huge improvement in the available fundamental stellar data and practical definition of the optical reference frame.