  We prove that the classes of graph algebras, Exel-Laca algebras, and
ultragraph algebras coincide up to Morita equivalence. This result answers the
long-standing open question of whether every Exel-Laca algebra is Morita
equivalent to a graph algebra. Given an ultragraph G we construct a directed
graph E such that C*(G) is isomorphic to a full corner of C*(E). As
applications, we characterize real rank zero for ultragraph algebras and
describe quotients of ultragraph algebras by gauge-invariant ideals.
