Criteria are presented that ensure that all zeros of a discrete system are inside the unit disc when a continuous-time plant with a time delay is sampled by use of a zero-order hold. A necessary and sufficient condition developed in this paper is a natural extension of the results for the case with no time delays. This paper also derives sufficient conditions for stable discrete zeros in terms of the coefficients of a continuous-time transfer function. They guarantee that all the sampled zeros stay inside the unit disc as long as the sampling period is selected to satisfy a simple relation.