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Cosmological variation of the fine structure constant $\alpha$ due to theevolution of a spatially homogeneous ultra-light scalar field ($m \sim H_0$)during the matter and $\Lambda$ dominated eras is analyzed. Agreement of$\Delta \alpha/\alpha$ with the value suggested by recent observations ofquasar absorption lines is obtained by adjusting a single parameter, thecoupling of the scalar field to matter.  Asymptotically $\alpha(t)$ in this model goes to a constant value$\bar{\alpha} \approx \alpha_0$ in the early radiation and the late $\Lambda$dominated eras. The coupling of the scalar field to (nonrelativistic) matterdrives $\alpha$ slightly away from $\bar{\alpha}$ in the epochs when thedensity of matter is important.  Simultaneous agreement with the more restrictive bounds on the variation$|\Delta \alpha/\alpha|$ from the Oklo natural fission reactor and frommeteorite samples can be achieved if the mass of the scalar field is on theorder of 0.5--0.6 $H_\Lambda$, where $H_\Lambda = \Omega_\Lambda^{1/2} H_0$.  Depending on the scalar field mass, $\alpha$ may be slightly smaller orlarger than $\alpha_0$ at the times of big bang nucleosynthesis, the emissionof the cosmic microwave background, the formation of early solar systemmeteorites, and the Oklo reactor. The effects on the evolution of $\alpha$ dueto nonzero mass for the scalar field are emphasized.  An order of magnitude improvement in the laboratory technique could lead to adetection of $(\dot{\alpha}/\alpha)_0$.Comment: 22 pages, 15 figures Version 2: The Oklo event is now considered as  localized in time. The initial conditions for the scalar field have been  refined. The numbers in the Table have been recomputed. Added Ref

Cosmological variation of the fine structure constant from an ultralight scalar field: The effects of mass