Summary: The artificial boundary method is applied to solve three-dimensional exterior problems. Two kinds of rotating ellipsoids are chosen as the artificial boundaries and the exact artificial boundary conditions are derived explicitly in terms of an infinite series. Then, the well-posedness of the coupled variational problem is obtained. It is found that the error estimates derived depend on the mesh size, the truncation term and the location of the artificial boundary. Three numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.