We study the supervisory control of dense-time discrete event systems (DESs) in which the supervisors employ finite-precision digital-clocks to observe the event occurrence times, thereby relaxing the assumption of the prior works that time can be measured precisely. In our paper, the passing of time is measured using the number of ticks generated by a digital-clock and we allow the plant events and digital-clock ticks to occur concurrently. We formalize the notion of a control policy that issues the control actions based on the observations of events and their occurrence times as measured using a digital-clock, and show that such a control policy can be equivalently represented as a "digitalized"-automaton, namely, an untimed-automaton that evolves over events (of the plant) and ticks (of the digital-clock). We introduce the notion of observability with respect to the partial observation of time resulting from the use of a digital-clock, and show that this property together with controllability serves as a necessary and sufficient condition for the existence of a supervisor to enforce a real-time specification on a dense-time discrete event plant. The observability condition presented in the paper is very different from the one arising due to a partial observation of events since a partial observation of time is in general nondeterministic (the number of ticks generated in any time interval can vary from execution to execution of a digital-clock). We also present a method to verify the proposed observability and controllability conditions, and an algorithm to compute a supervisor when such conditions are satisfied.