In a recent paper we have proposed terminological default logic as a formalism which combines both means for structured representation of classes and objects, and for default inheritance of properties. The major drawback which terminological default logic inherits from general default logic is that it does not take precedence of more speciic defaults over more general ones into account. This behaviour has already been criticized in the general context of default logic, but it is all the more problematic in the terminological case where the emphasis lies on the hierarchical organization of concepts. The present paper addresses the problem of modifying terminologi-cal default logic such that more speciic defaults are preferred. It turns out that the existing approaches for expressing priorities between defaults do not seem to be appropriate for this purpose. Therefore we shall consider an alternative approach for dealing with prioritization in the framework of Reiter's default logic. The formalism is presented in the general setting of default logic where priorities are given by an arbitrary partial ordering on the defaults. We shall exhibit some interesting properties of the new formalism, compare it with existing approaches, and describe an algorithm for computing extensions.