We develop a functional-discrete method with a high order of accuracy to find a numerical solution of an eigenvalue transmission problem. It allows to approximate the trial eigenvalue with any desired accuracy. This approach has no restriction on the number of eigenvalues, an approximation to which can be found. The convergence rate is proved as in the case of the geometric series. It is shown that depending on the data of the original problem, two kinds of eigenvalue sequences may exist. For the first one, the convergence rate increases as the ordinal number of the trial eigenvalue increases. For the second one, the convergence rate is the same for all eigenvalues and does not depend on the ordinal number of the trial eigenvalue. Based on the asymptotic behavior of the eigenvalues of the basic problem and the functional-discrete method, a qualitative result on the arrangement of eigenvalues of the original problem is established. A number of numerical examples are given to support the theory.