We have recorded the R(0)nu(CO) = 1 <- 0 IR spectrum of CO and its isotopomers in superfluid helium nanodroplets. For droplets with average size N greater than or similar to 2000 helium atoms, the transition exhibits a Lorentzian shaped linewidth of 0.034 cm(-1), indicating a homogeneous broadening mechanism. The rotational constants could be deduced and were found to be reduced to about 60% of the corresponding gas-phase values (63% for the reference C-12 O-16 species). Accompanying calculations of the pure rotational spectra were carried out using the method of correlated basis functions in combination with diffusion Monte Carlo (CBF/DMC). These calculations show that both the reduction of the rotational B constant and the line broadening can be attributed to phonon-rotation coupling. The reduction in B is confirmed by path integral correlation function calculations for a cluster of 64 He-4 atoms, which also reveal a non-negligible effect of finite size on the collective modes. The phonon-rotation coupling strength is seen to depend strongly on the strength and anisotropy of the molecule-helium interaction potential. Comparison with other light rotors shows that this coupling is particularly high for CO. The CBF/DMC analysis shows that the J = 1 rotational state couples effectively to phonon states, which are only present in large helium droplets or bulk. In particular, they are not present in small clusters with n <= 20, thereby accounting for the much narrower linewidths and larger B constant measured for these sizes