  We prove that a graph C*-algebra with exactly one proper nontrivial ideal is
classified up to stable isomorphism by its associated six-term exact sequence
in K-theory. We prove that a similar classification also holds for a graph
C*-algebra with a largest proper ideal that is an AF-algebra. Our results are
based on a general method developed by the first named author with Restorff and
Ruiz. As a key step in the argument, we show how to produce stability for
certain full hereditary subalgebras associated to such graph C*-algebras. We
further prove that, except under trivial circumstances, a unique proper
nontrivial ideal in a graph C*-algebra is stable.
