Computing K-theory and Ext for graph C^*-algebras

K-theory and Ext are computed for the C^*-algebra C^*(E) of any countable directed graph E. The results generalize the K-theory computations of Raeburn and Szymański and the Ext computations of Tomforde for row-finite graphs. As a consequence, it is shown that if A is a countable \0,1\ matrix and EA is the graph obtained by viewing A as a vertex matrix, then C^*(EA) is not necessarily Morita equivalent to the Exel-Laca algebra OA.