The pure rotational transition (J, j, l) = (101)<--(000) of the weakly bound CO-He complex has been observed in the 17 GHz region for five isotopomers, namely (CO)-C-12-O-16-He-4, (CO)-C-13-O-16-He-4, (CO)-C-12-O-18-He-4, (CO)-C-13-O-18-He-4, and (CO)-C-13-O-17-He-4, using a pulsed-jet cavity Fourier-transform microwave spectrometer. Hyperfine structure due to the quadrupolar O-17 (I = 5/2) nucleus has been observed and analyzed to yield the quadrupole coupling parameters. A new microwave-terahertz double resonance spectrometer has been used to carry out an experiment on (CO)-C-13-O-16-He-4 in which a submillimeter-wave pump transition (J, j, l) = (110)<--(000) and a microwave signal transition (211) <--(110) were detected. Infrared spectra of (CO)-C-13-O-16-He-4 and (CO)-C-12-O-18-He-4, have been obtained in the 2100 cm(-1) region of the C-O stretch using a tunable diode laser spectrometer and a long-path (200 m), low-temperature (46 K) equilibrium gas cell. The combined data are sufficient to construct essentially complete experimental energy level schemes for the bound states of the (CO)-C-12-O-16-He-4, (CO)-C-13-O-16-He-4, and (CO)-C-12-O-18-He-4 forms of the complex. These energies are compared with calculated levels derived from two intermolecular potential models, V-(3,V-3,V-3) and XC (fit). It is shown that the new spectra, in particular those of (CO)-C-13-O-16-He-4, discriminate between the two models and indicate that V-(3,V-3,V-3) provides a better representation of the potential in the region of the attractive well probed by the bound states. A quasibound state of CO-He was observed for the first time, and its width due to predissociation was approximately determined, providing a sensitive measure of the depth of the potential.