  In our recent papers the centralizer construction was applied to the series
of classical Lie algebras to produce the quantum algebras called (twisted)
Yangians. Here we extend this construction to the series of the symmetric
groups S(n). We study the `stable' properties of the centralizers of S(n-m) in
the group algebra C[S(n)] as n increases with m fixed. We construct a limit
centralizer algebra A and describe its algebraic structure. The algebra A turns
out to be closely related with the degenerate affine Hecke algebras. We also
show that the so-called tame representations of the infinite symmetric group
yield a class of natural A-modules.
