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We present an analysis of the broad absorption line (BAL) velocity shift that appeared in one of the outflow systems in quasar SDSS J1042+1646. Observations were taken by the Hubble Space Telescope/Cosmic Origin Spectrograph in 2011 and 2017 in the 500 -- 1050 $Å$ rest frame. The outflow's velocity centroid shifted by $\sim$ --1550 km s$^{-1}$ from --19,500 km s$^{-1}$ to --21,050 km s$^{-1}$ over a rest-frame time of 3.2 yr. The velocity shift signatures are most apparent in the absorption features from the Ne VIII $λλ$ 770.41, 780.32 doublet and are supported by the absorption troughs from OV $λ$ 629.73 and the Mg X $λλ$ 609.79, 624.94 doublet. This is the first time where a quasar outflow velocity shift is observed in troughs from more than one ion and in distinct troughs from a doublet transition (Ne VIII). We attribute the velocity shift to an acceleration of an existing outflow as we are able to exclude photoionization changes and motion of material into and out of the line of sight as alternate explanations. This leads to an average acceleration of 480 km s$^{-1}$ yr$^{-1}$ (1.52 cm s$^{-2}$) in the quasar rest frame. Both the acceleration and the absolute velocity shift are the largest reported for a quasar outflow to date. Based on the absorption troughs of the O V* multiplet, we derive a range for the distance of the outflow ($R$) from the central source, 0.05 pc $<$ $R$ $<$ 54.3 pc. This outflow shows similarities with the fast X-ray outflow detected in quasar PG 1211+143. We use the acceleration and velocity shift to constrain radiatively accelerated active galactic nucleus disk-wind models and use them to make predictions for future observations.