Hipparcos parallaxes fix distances to individual stars in the Hyades cluster with an accuracy of 6%. We use the Hipparcos (and Tycho-2) 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. The improved parallaxes as a set are statistically consistent with the Hipparcos parallaxes. 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. The colour-absolute magnitude diagram of the cluster based on the new paral- laxes shows a well-defined main sequence with two gaps/turn-offs. These features provide the first direct observational support of Boehm-Vitenses prediction that (the onset of) surface convection in stars affects their B-V colours. We present and discuss the theoretical HRD for an objectively defined set of 88 high-fidelity members of the cluster as well as the delta Scuti star theta^2 Tau, the giants delta^1, theta^1, epsilon, and gamma 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 members in the Hertz- sprung-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 theoretical stellar evolutionary modelling, tailored to the chemical composition and age of the Hyades, over the large stellar mass range probed by Hipparcos.
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