  We extend the classical Schur-Weyl duality between representations of the
groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite
symmetric group $\sinf$. Our construction is based on a "dynamic," or
inductive, scheme of Schur-Weyl dualities. It leads to a new class of
representations of the infinite symmetric group, which have not appeared
earlier. We describe these representations and, in particular, find their
spectral types with respect to the Gelfand-Tsetlin algebra.
  The main example of such a representation acts in an incomplete infinite
tensor product. As an important application, we consider the weak limit of the
so-called Coxeter-Laplace operator, which is essentially the Hamiltonian of the
XXX Heisenberg model, in these representations.
