Understanding many physical processes in the solar atmosphere requires determination of the magnetic field in each atmospheric layer. However, direct measurements of the magnetic field in the Sun's corona are difficult to obtain. Using observations with the Corona! Multi-channel Polarimeter, we have determined the spatial distribution of the plasma density in the corona and the phase speed of the prevailing transverse magnetohydrodynamic waves within the plasma. We combined these measurements to map the plane-of-sky component of the global coronal magnetic field. The derived field strengths in the corona, from 1.05 to 1.35 solar radii, are mostly 1 to 4 gauss. Our results demonstrate the capability of imaging spectroscopy in coronal magnetic field diagnostics.