Solar-type protostars have been shown to harbor highly deuterated complex organics, as evidenced, for instance, by the high relative abundances of doubly and triply deuterated isotopologs. While this degree of deuteration may provide important clues in studying the formation of these species, spectroscopic information on multiply deuterated isotopologs is often insufficient. In particular, searches for triply deuterated methanol, CD3OH, are hampered to a large extent by the lack of intensity information from a spectroscopic model. The aim of the present study is to develop a spectroscopic model of CD3OH in low-lying torsional states that is sufficiently accurate to facilitate further searches for CD3OH in space. We performed a new measurement campaign for CD3OH involving three spectroscopic laboratories that covers the 34 GHz-1.1 THz and the 20-900 cm-1 ranges. The analysis was performed using the torsion-rotation Hamiltonian model based on the rho-axis method. We determined a model that describes the ground and first excited torsional states of CD3OH, up to quantum numbers J < 55 and Ka ≤ 23, and we derived a line list for radio-astronomical observations. The resulting line list is accurate up to at least 1.1 THz and should be sufficient for all types of radio-astronomical searches for this methanol isotopolog. This line list was used to search for CD3OH in data from the Protostellar Interferometric Line Survey of IRAS 16293-2422 using the Atacama Large Millimeter/submillimeter Array. Specifically, CD3OH is securely detected in the data, with a large number of clearly separated and well-reproduced lines. We not only detected lines belonging to the ground torsional state, but also several belonging to the first excited torsional state. The derived column density of CD3OH and abundance relative to the non-deuterated isotopolog confirm the significant enhancement of this multiply deuterated variant. This finding is in line with other observations of multiply deuterated complex organic molecules and may serve as an important constraint on their formation models.