  In 1989, Dicks and Dunwoody proved the Almost Stability Theorem, which has
among its corollaries the Stallings-Swan theorem that groups of cohomological
dimension one are free. In this article, we use a nestedness result of Bergman,
Bowditch, and Dunwoody to simplify somewhat the proof of the finitely generable
case of the Almost Stability Theorem. We also simplify the proof of the non
finitely generable case.
  The proof we give here of the Almost Stability Theorem is essentially self
contained, except that in the non finitely generable case we refer the reader
to the original argument for the proofs of two technical lemmas about groups
acting on trees.
