We present orbits for 22 spectroscopic binaries in the field of the old open cluster M67. All but three are certain cluster members on the basis of both radial-velocity and proper-motion evidence. Five of the member binaries are double lined; one of these is a triple system with a single-lined binary and a tertiary with no detected velocity variation. The observed frequency of binaries with periods less than 1000 days among the solar-mass M67 members is 18%. However, after correction for the imposition of a magnitude limit on the sample, the M67 short-period binary frequency is found to be between 9% and 15%. While such a frequency is somewhat lower than the binary frequency among field solar-mass stars, the two are not distinguishable at a significant level. We have found a transition from circular to eccentric orbits among the main-sequence binaries at a period between 10.3 and 11.0 days, longer than similar transition periods found in younger binary populations and shorter than that found in the galactic halo. An increase of transition period with age is consistent with the interpretation that the shorter-period orbits have been tidally circularized. Circular orbits are found at longer periods among the cluster giants. With the exception of the giant binary S1221, the eccentricity distribution along the giant branch is consistent with tidal circularization processes being more effective at larger stellar radii and deeper convection zones. Three binaries in the cluster halo have orbital eccentricities among the highest in the cluster, which would be consistent with their ejection from the cluster core by stellar encounters. The giant binary S1221 is notable in having a circular or very-near-circular orbit at a period of 6450 days; it has also been noted to have a CN-enriched surface composition.