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We have used high-precision differential astrometry from the Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES) project and radial velocity measurements covering a time span of 20 years to determine the orbital parameters of the 88 Tau A system. 88 Tau is a complex hierarchical multiple system comprising a total of six stars; we have studied the brightest four, consisting of two short-period pairs orbiting each other with an ~18 yr period. We present the first orbital solution for one of the short-period pairs, and determine the masses of the components and distance to the system to the level of a few percent. In addition, our astrometric measurements allow us to make the first determination of the mutual inclinations of the orbits. We find that the subsystems are not coplanar.