  Let K be an abstract elementary class satisfying the joint embedding and the
amalgamation properties. Let m be a cardinal above the the L\"owenheim-Skolem
number of the class. Suppose K satisfies the disjoint amalgamation property for
limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian
Pairs, does not have long splitting chains, and satisfies locality of
splitting, for the precise description of long splitting chains and locality},
then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M.
  This theorem extends results of Shelah, Kolman and Shelah, and Shelah and
Villaveces. A preliminary version of our uniqueness theorem was used by
Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for
tame abstract elementary classes.
