Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.