We have constructed an instrument to measure the polarization of light emitted by the solar corona in order to constrain the strength and orientation of coronal magnetic fields. We call this instrument the Coronal Multichannel Polarimeter (CoMP). The CoMP is integrated into the Coronal One Shot coronagraph at Sacramento Peak Observatory and employs a combination birefringent filter and polarimeter to form images in two wavelengths simultaneously over a 2.8R⊙ field of view. The CoMP measures the complete polarization state at the 1074.7 and 1079.8 Fe XIII coronal emission lines, and the 1083.0 nm He I chromospheric line. In this paper we present design drivers for the instrument, provide a detailed description of the instrument, describe the calibration methodology, and present some sample data along with estimates of the uncertainty of the measured magnetic field.