Signature-based abduction aims at building hypotheses over aspecified set of names, the signature, that explain an observa-tion relative to some background knowledge. This type of ab-duction is useful for tasks such as diagnosis, where the vocab-ulary used for observed symptoms differs from the vocabu-lary expected to explain those symptoms. We present the firstcomplete method solving signature-based abduction for ob-servations expressed in the expressive description logicALC,which can include TBox and ABox axioms. The method isguaranteed to compute a finite and complete set of hypothe-ses, and is evaluated on a set of realistic knowledge bases.