We have obtained accurate values for the higher electric multipole moments of CO. We rely on finite-field coupled cluster calculations with large gaussian-type basis sets. At the CCSD(T) level of theory the quadrupole moment varies as [Theta (R) - Theta (R-e)]/ea(0)(2) = 1.03(R - R-e) + 0.40(R - R-e)(2) around the experimental bond length of R-e = 2.132221 a(0). For the octopole and hexadecapole moment we find [Omega (R-e)]/ea(0)(3 =) 1.49(R - R-e) - 1.81(R - R-e)(2) - 0.45(R - R-e)(3) -0.17(R - R-e)(4) (R - R-e)(4) , [Phi)(R) - Phi (R-e)]/ea(0)(4) = -4.61(R - R-e) + 0.67(R - Re)(2) + 2.29(R - R-e)(3) + 1.04(R - R-e)(4). Our best values Omega = 3.46 ea(0)(3) and Phi = -9.07 ea(0)(4) are in reasonable agreement with recent experimental estimates deduced from farinfrared absorption in the high-density gas spectrum of CO-Ar. Agreement is less satisfactory with values obtained from the liquid spectrum.