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We present new absolute proper motion measurements for the Arches and Quintuplet clusters, two young massive star clusters near the Galactic center. Using multi-epoch HST observations, we construct proper motion catalogs for the Arches ($\sim$35,000 stars) and Quintuplet ($\sim$40,000 stars) fields in ICRF coordinates established using stars in common with the Gaia EDR3 catalog. The bulk proper motions of the clusters are measured to be ($μ_{α*}$, $μ_δ$) = (-0.80 $\pm$ 0.032, -1.89 $\pm$ 0.021) mas/yr for the Arches and ($μ_{α*}$, $μ_δ$) = (-0.96 $\pm$ 0.032, -2.29 $\pm$ 0.023) mas/yr for the Quintuplet, achieving $\sim$5x higher precision than past measurements. We place the first constraints on the properties of the cluster orbits that incorporate the uncertainty in their current line-of-sight distances. The clusters will not approach closer than $\sim$25 pc to SgrA*, making it unlikely that they will inspiral into the Nuclear Star Cluster within their lifetime. Further, the cluster orbits are not consistent with being circular; the average value of r$_{apo}$ / r$_{peri}$ is $\sim$1.9 (equivalent to eccentricity of $\sim$0.31) for both clusters. Lastly, we find that the clusters do not share a common orbit, challenging one proposed formation scenario in which the clusters formed from molecular clouds on the open stream orbit derived by Kruijssen et al. (2015). Meanwhile, our constraints on the birth location and velocity of the clusters offer mild support for a scenario in which the clusters formed via collisions between gas clouds on the x1 and x2 bar orbit families.