This paper investigates the BIBO stability of a class of fractional-delay systems with rational orders, a stability that holds true if all the characteristic roots have negative real parts only. Based on the Argument Principle for complex functions as well as Hassard's technique for ordinary time-delay systems, an explicit formula is established for calculating the number of characteristic roots lying in the closed right-half complex plane of the first sheet of the Riemann surface, and in turn a sufficient and necessary condition is obtained for testing the BIBO stability of fractional-delay systems. As shown in the illustrative examples, this stability criterion involves easy computation and works effectively. (C) 2011 Elsevier Ltd. All rights reserved.