We present absolute trigonometric parallaxes and relative proper motions for three members of the Pleiades, obtained with the Hubble Space Telescope's Fine Guidance Sensor 1r, a white-light interferometer. We estimate spectral types and luminosity classes of the stars comprising the astrometric reference frame from R 2000 spectra, VJHK photometry, and reduced proper motions. From these we derive estimates of absolute parallaxes and introduce them into our model as observations with error. We constrain the three cluster members to have a 1 s dispersion in distance less than 6.4 pc and find an average pabs = 7.43 ± 0.17 ± 0.20 mas, where the second error is systematic due to member placement within the cluster. This parallax corresponds to a distance of 134.6 ± 3.1 pc or a distance modulus of m - M = 5.65 ± 0.05 for these three Pleiades stars, presuming a central location. This result agrees with three other independent determinations of the Pleiades distance. Presuming that the cluster depth systematic error can be significantly reduced because of the random placement of these many members within the cluster, these four independent measures yield a best-estimate Pleiades distance of pabs = 7.49 ± 0.07 mas, corresponding to a distance of 133.5 ± 1.2 pc or a distance modulus of m - M = 5.63 ± 0.02. This resolves the dispute between the main-sequence fitting and the Hipparcos distance moduli in favor of main-sequence fitting.