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From numerous radial velocities as well as Johnson B and V differential photoelectric photometry, we have determined the orbital elements and other properties of the single-lined triple system HD 126516. This system consists of a narrow-lined F5 V star and an unseen M dwarf companion in a 2.1241 day circular orbit. The small, low-mass secondary produces detectable eclipses of the primary, and that pair has been given the variable star name V349 Vir. Variations of the center-of-mass velocity of this short-period system have an orbital period of 702.7 days or 1.92 yr and an eccentricity of 0.36. The third star is likely a K or M dwarf. From an analysis of our photometry, we conclude that the primary of HD 126516 is not a γ Dor variable. Comparison with evolutionary tracks indicates that the primary is slightly metal-poor and has an age of 2.5 Gyr. The projected rotational velocity of the primary is very low, just 4 km s−1, which is 10 times less than its synchronous rotational velocity. Thus, either that component's rotation is extremely non-synchronous or the inclinations of the rotational and orbital axes are very different, and so the primary has a very large spin–orbit misalignment. Because of the moderate age of the system and the fact that its orbit is already circularized, neither situation is expected theoretically.