We examine the late-time (nucleosynthesis and later) cosmological implications of brane-world scenarios having large (millimeter sized) extra dimensions. In particular, recent proposals for understanding why the extra dimensions are so large in these models indicate that moduli like the radion appear (to four-dimensional observers) to be extremely light, with a mass of order 10^{-33} eV, allowing them to play the role of the light scalar of quintessence models. The radion-as-quintessence solves a long-standing problem since its small mass is technically natural, in that it is stable against radiative corrections. Its challenges are to explain why such a light particle has not been seen in precision tests of gravity, and why Newton's constant has not appreciably evolved since nucleosynthesis. We find the couplings suggested by stabilization models can provide explanations for both of these questions. We identify the features which must be required of any earlier epochs of cosmology in order for these explanations to hold.

11 Pages, LaTeX, 4 figures