We describe the Submillimeter Array (SMA) Polarimeter, a polarization converter and feed multiplexer installed on the SMA. The Polarimeter uses narrow-band quarter-wave plates to generate circular polarization sensitivity from the linearly-polarized SMA feeds. The wave plates are mounted in rotation stages under computer control so that the polarization handedness of each antenna is rapidly selectable. Positioning of the wave plates is found to be highly repeatable, better than 0.2 degrees. Although only a single polarization is detected at any time, all four cross correlations of left- and right-circular polarization are efficiently sampled on each baseline through coordinated switching of the antenna polarizations in Walsh function patterns. The initial set of anti-reflection coated quartz and sapphire wave plates allows polarimetry near 345 GHz; these plates have been have been used in observations between 325 and 350 GHz. The frequency-dependent cross-polarization of each antenna, largely due to the variation with frequency of the retardation phase of the single-element wave plates, can be measured precisely through observations of bright point sources. Such measurements indicate that the cross-polarization of each antenna is a few percent or smaller and stable, consistent with the expected frequency dependence and very small alignment errors. The Polarimeter is now available for general use as a facility instrument of the SMA.