It is shown that the outer automorphism group of a Coxeter group W of
finite rank is finite if the Coxeter graph contains no infinite bonds.
A key step in the proof is to show that if the group is irreducible a
nd Pi(1) and Pi(2) any two bases of the root system of W, then Pi(2) =
+/-w Pi(1) for some w is an element of W. The proof of this latter fa
ct employs some properties of the dominance order on the root system i
ntroduced by Brink and Howlett.