Solar-type prestellar cores and protostars display frequently large amounts of deuterated organic molecules and in particular high relative abundances of doubly and triply deuterated isotopologs. Recent findings on CHD_2_OH and CD_3_OH toward IRAS 16293-2422 suggest that even fully deuterated methanol, CD_3_OD, may be detectable as well. However, searches for CD_3_OD are hampered in particular by the lack of intensity information from a spectroscopic model. The objective of the present investigation is to develop a spectroscopic model of CD_3_OD in low-lying torsional states that is sufficiently accurate to facilitate searches for this isotopolog in space. We carried out a new measurement campaign for CD_3_OD involving two spectroscopic laboratories that covers the 34 GHz-1.1THz range. A torsion-rotation Hamiltonian model based on the rho-axis method was employed for our analysis. Our resulting model describes the ground and first excited torsional states of CD_3_OD well up to quantum numbers J<=51 and Ka<=23. We derived a line list for radio-astronomical observations from this model that is accurate up to at least 1.1THz and should be sufficient for all types of radio-astronomical searches for this methanol isotopolog. This line list was used to search for CD_3_OD in data from the Protostellar Interferometric Line Survey of IRAS 16293-2422 obtained with the Atacama Large Millimeter/ submillimeter Array. While we found several emission features that can be attributed largely to CD_3_OD, their number is as yet insufficient to establish a clear detection. Nevertheless, the estimate of 2x10^15cm^-2^ derived for the CD3OD column density may be viewed as an upper limit that can be compared to column densities of CD_3_OH, CH_3_OD, and CH_3_OH. The comparison indicates that the CD_3_OD column density toward IRAS 16293-2422 is in line with the enhanced D/H ratios observed for multiply deuterated complex organic molecules.