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We have obtained near-infrared adaptive optics imaging and collected additional radial velocity observations to search for a third component in the extremely dusty short-period binary system BD +20°307. Our image shows no evidence for a third component at separations greater than 19 AU. Our four seasons of radial velocities have a constant center-of-mass velocity and are consistent with the systemic velocities determined at two earlier epochs. Thus, the radial velocities also provide no support for a third component. Unfortunately, the separation domains covered by our imaging and radial velocity results do not overlap. Thus, we examined the parameters for possible orbits of a third component that could have been missed by our current observations. With our velocities we determined improved circular orbital elements for the 3.4 day double-lined binary. We also performed a spectroscopic abundance analysis of the short-period binary components and conclude that the stars are a mid- and a late-F dwarf. We find that the iron abundances of both components, [Fe/H] = 0.15, are somewhat greater than the solar value and comparable to that of stars in the Hyades. Despite the similarity of the binary components, the lithium abundances of the two stars are very unequal. The primary has log epsilon (Li) = 2.72, while in the secondary log epsilon (Li) ⩽1.46, which corresponds to a difference of at least a factor of 18. The very disparate lithium abundances in very similar stars make it impossible to ascribe a single age to them. While the system is likely at least 1 Gyr old, it may well be as old as the Sun.