  We identify a regime of the AdS/CFT correspondence in which we can
quantitatively match N=4 super Yang-Mills (SYM) for small 't Hooft coupling
with weakly coupled type IIB string theory on AdS_5 x S^5. We approach this
regime by taking the same decoupling limit on both sides of the correspondence.
On the gauge theory side only the states in the SU(2) sector survive, and in
the planar limit the Hamiltonian is given by the XXX_{1/2} Heisenberg spin
chain. On the string theory side we show that the decoupling limit corresponds
to a non-relativistic limit. In this limit some of the bosonic modes and all of
the fermionic modes of the string become infinitely heavy and decouple. We
first take the decoupling limit of the string sigma-model classically. This
enables us to identify a semi-classical regime with semi-classical string
states even though we are in a regime corresponding to small 't Hooft coupling.
We furthermore analyze the quantum corrections that enter in taking the limit.
  From this we infer that gauge theory and string theory match, both in terms
of the action and the spectrum, for the leading part and the first correction
away from the semi-classical regime. Finally we consider the implications for
the hitherto unexplained matching of the one-loop contribution to the energy of
certain gauge theory and string theory states, and we explain how our results
give a firm basis for the matching of the Hagedorn temperature in
hep-th/0608115.
