The motion of the components of a close binary system near a very massive body is investigated within the framework of the three-body problem. Cases requiring consideration of relativistic effects are also examined. It is shown that the close binary system often breaks up at some distance from the very massive body on approaching it. In many cases, the breakup of the close binary system is preceded by an abrupt decrease in the separation of its components. Possible consequences of close binary system breakup for the abundance of such systems in galaxies are discussed briefly.