  For a directed graph $E$, we compute the $K$-theory of the $C^*$-algebra
$C^*(E)$ from the Cuntz-Krieger generators and relations. First we compute the
$K$-theory of the crossed product $C^*(E)\times_\gamma\IT$, and then using
duality and the Pimsner-Voiculescu exact sequence we compute the $K$-theory of
$C^*(E)\otimes\CK \cong (C^*(E)\times\IT)\times\IZ$. The method relies on the
decomposition of $C^*(E)$ as an inductive limit of Toeplitz graph
$C^*$-algebras, indexed by the finite subgraphs of $E$. The proof and result
require no special asssumptions about the graph, and is given in
graph-theoretic terms. This can be helpful if the graph is described by
pictures rather than by a matrix.
