We present a multi-epoch Hubble Space Telescope (HST) study of stellar proper motions (PMs) for four fields spanning 200 degrees along the Sagittarius (Sgr) stream: one trailing arm field, one field near the Sgr dwarf spheroidal tidal radius, and two leading arm fields. We determine absolute PMs of dozens of individual stars per field, using established techniques that use distant background galaxies as stationary reference frame. Stream stars are identified based on combined color-magnitude diagram and PM information. The results are broadly consistent with the few existing PM measurements for the Sgr galaxy and the trailing arm. However, our new results provide the highest PM accuracy for the stream to date, the first PM measurements for the leading arm, and the first PM measurements for individual stream stars; we also serendipitously determine the PM of the globular cluster NGC~6652. In the trailing-arm field, the individual PMs allow us to kinematically separate trailing-arm stars from leading-arm stars that are 360 degrees further ahead in their orbit. Also, in three of our fields we find indications that two distinct kinematical components may exist within the same arm and wrap of the stream. Qualitative comparison of the HST data to the predictions of the Law & Majewski and Penarrubia et al. N-body models show that the PM measurements closely follow the predicted trend with Sgr longitude. This provides a successful consistency check on the PM measurements, as well as on these N-body approaches (which were not tailored to fit any PM data).
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