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There are indications from the study of quasar absorption spectra that thefine structure constant $\alpha$ may have been measurably smaller for redshifts$z>2.$ Analyses of other data ($^{149}$Sm fission rate for the Oklo naturalreactor, variation of $^{187}$Re $\beta$-decay rate in meteorite studies,atomic clock measurements) which probe variations of $\alpha$ in the morerecent past imply much smaller deviations from its present value. In this workwe tie the variation of $\alpha$ to the evolution of the quintessence fieldproposed by Albrecht and Skordis, and show that agreement with all these data,as well as consistency with WMAP observations, can be achieved for a range ofparameters. Some definite predictions follow for upcoming space missionssearching for violations of the equivalence principle.Comment: Final version, to be published in Phys Rev

Time variation of the fine structure constant driven by quintessence