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We present the results of an investigation of a highly variable C iv broad absorption line (BAL) feature in spectra of the quasar SBS 1408+544 ( z = 2.337) that shows a significant shift in velocity over time. This source was observed as a part of the Sloan Digital Sky Survey (SDSS) Reverberation Mapping project and the SDSS-V Black Hole Mapper Reverberation Mapping project, and has been included in two previous studies, both of which identified significant variability in a high-velocity C iv BAL on timescales of just a few days in the quasar rest frame. Using ∼130 spectra acquired over 8 yr of spectroscopic monitoring with SDSS, we have determined that this BAL is not only varying in strength, but is also systematically shifting to higher velocities. Using cross-correlation methods, we measure the velocity shifts (and corresponding acceleration) of the BAL over a wide range of timescales, measuring an overall velocity shift of ${\rm{\Delta }}v=-{683}_{-84}^{+89}$ km s ^−1 over the 8 yr monitoring period. This corresponds to an average rest-frame acceleration of a = 1.04 ${}_{-0.13}^{+0.14}$ cm s ^−2 , though the magnitude of the acceleration on shorter timescales is not constant throughout. We place our measurements in the context of BAL-acceleration models and examine various possible causes of the observed velocity shift.