We develop a formalism describing Coulomb excitation in relativistic heavy-ion collisions as a general coupled-channel problem. The Schrödinger equation for the target nucleus is expressed as a set of coupled integral equations for a [Formula Presented] matrix, whose on-shell matrix elements give the excitation amplitudes. The method of Kowalsky and Noyes is used to regularize the kernels of these integral equations, making the equations amenable to numerical solution. An application of the method is made to the study of the population of a model state representing an [Formula Presented] giant resonance in [Formula Presented] at an excitation energy of 20 MeV, due to the electromagnetic field of a [Formula Presented] projectile with kinetic energy per nucleon up to 100 GeV. The results of the numerical solution of the regularized coupled integral equations are compared to results obtained from the Born series, and from the long-wavelength approximation. These approximations are found to be adequate up to bombarding energies per nucleon of 3 GeV. We investigate the reasons for failure of these approximations at higher bombarding energy. Due to the finite travel time of the electromagnetic pulse across the target nucleus, the sudden approximation is found to be inapplicable to the excitation of the giant resonance. Thus the method of numerical solution of the regularized coupled integral equations seems to be the most suitable approach to the study of very high-energy Coulomb excitation of giant resonances.