This paper deals with the admissibility analysis of continuous-time descriptor systems with a constant delay. A transformation-based model is used to convert the original system into a delay-free system. As a result, new regularity and impulse-free characteristics, depending on a constant-delay, have been presented. Based on the generalized Lyapunov function, a necessary and sufficient condition has been presented to ensure that the delay-descriptor system is asymptotically stable. The obtained conditions are formulated as a set of linear matrix inequalities. Finally, an example is presented to test the merit and validity of the proposed method.