In this note, we give new stability tests which enable one to fully characterize the H-infinity-stability of systems with transfer function G(s) = r(s)/p(s)+q(s)e(-sh), where h > 0 and p, q, r are real polynomials in the variable s(mu) for 0 < mu < 1. As an application of this, in the case r(s) = 1 and deg p = deg q = 1, families of H-infinity-stabilizing controllers are given and a complete parametrization of all H-infinity-stabilizing controllers is obtained when vertical bar lim(s is an element of c\R-vertical bar s vertical bar ->infinity) p(s)\q(s)vertical bar > 1. (c) 2007 Elsevier Ltd. All rights reserved.