We have undertaken a spectroscopic search for ultra compact dwarf galaxies (UCDs) in the dense core of the dynamically evolved, massive Coma cluster as part of the HST/ACS Coma Cluster Treasury Survey. UCD candidates were initially chosen based on color, magnitude, degree of resolution within the ACS images, and the known properties of Fornax and Virgo UCDs. Follow-up spectroscopy with Keck/LRIS confirmed 27 candidates as members of the Coma Cluster, a success rate > 60% for targeted objects brighter than M_R = -12. Another 14 candidates may also prove to be Coma members, but low signal-to-noise spectra prevent definitive conclusions. An investigation of the properties and distribution of the Coma UCDs finds these objects to be very similar to UCDs discovered in other environments. The Coma UCDs tend to be clustered around giant galaxies in the cluster core and have colors/metallicity that correlate with the host galaxy. With properties and a distribution similar to that of the Coma cluster globular cluster population, we find strong support for a star cluster origin for the majority of the Coma UCDs. However, a few UCDs appear to have stellar population or structural properties which differentiate them from the old star cluster populations found in the Coma cluster, perhaps indicating that UCDs may form through multiple formation channels.