A generalization is given of the problem on the impression of a circular stamp when the elastic stamp makes contact with an unbounded elastic layer. Application of the Hankel integral transform in the region of the layer and the properties of generalized orthogonality of eigenfunctions in the region of the circular cylinder (stamp) permits reducing the problem to an infinite system of linear algebraic equations admitting of effective solution by the truncation method. The classical problem of a rigid stamp impressed into an elastic half-space has been subjected to generalization in several directions in recent decades. Thus, the impression of a rigid stamp into an elastic layer has been considered in a number of papers (see [1], for example), on the other hand, the problem of the contact between an elastic cylinder and a half-space has been studied in [2]. Finally, the even more general problem on the impression of an elastic cylinder into an elastic layer has been investigated in [3]. The problem has been reduced to infinite linear algebraic systems, which are effectively solvable for sufficiently thick layers, since the solution is expanded in power series of a small parameter, the ratio of the cylinder radius to the layer thickness. The same problem is reduced in this paper to infinite systems of a different kind, which are suitable for any ratios between the geometric parameters. Results of numerical computations of the stiffness of the system under consideration are presented.