Abstract. Hipparcos parallaxes fix distances to individual stars in the Hyades cluster with an accuracy of ∼6 percent. We use the Hipparcos proper motions, which have a larger relative precision than the trigonometric parallaxes, to derive ∼3 times more precise distance estimates, by assuming that all members share the same space motion. An investigation of the available kinematic data confirms that the Hyades velocity field does not contain significant structure in the form of rotation and/or shear, but is fully consistent with a common space motion plus a (one-dimensional) internal velocity dispersion of ∼0.30 km s −1 . The improved parallaxes as a set are statistically consistent with the Hipparcos parallaxes. The maximum expected systematic error in the proper motion-based parallaxes for stars in the outer regions of the cluster (i.e., beyond ∼2 tidal radii ∼20 pc) is < ∼ 0.30 mas. The new parallaxes confirm that the Hipparcos measurements are correlated on small angular scales, consistent with the limits specified in the Hipparcos Catalogue, though with significantly smaller "amplitudes" than claimed by Narayanan & Gould. We use the Tycho-2 long time-baseline astrometric catalogue to derive a set of independent proper motion-based parallaxes for the Hipparcos members. The new parallaxes provide a uniquely sharp view of the three-dimensional structure of the Hyades. The colour-absolute magnitude diagram of the cluster based on the new parallaxes shows a well-defined main sequence with two "gaps"/"turn-offs". These features provide the first direct observational support of Böhm-Vitense's prediction that (the onset of) surface convection in stars significantly affects their (B − V ) colours. We present and discuss the theoretical Hertzsprung-Russell diagram (log L versus log T eff ) for an objectively defined set of 88 high-fidelity members of the cluster as well as the δ Scuti star θ 2 Tau, the giants δ 1 , θ 1 , , and γ Tau, and the white dwarfs V471 Tau and HD 27483 (all of which are also members). The precision with which the new parallaxes place individual Hyades in the Hertzsprung-Russell diagram is limited by (systematic) uncertainties related to the transformations from observed colours and absolute magnitudes to effective temperatures and luminosities. The new parallaxes provide stringent constraints on the calibration of such transformations when combined with detailed theoretical stellar evolutionary modelling, tailored to the chemical composition and age of the Hyades, over the large stellar mass range of the cluster probed by Hipparcos.