A novel real-time discrete-event admission control (AC) scheme for high-speed networks is proposed with the aim of attaining a desired quality of service (QoS) and high network utilization. The AC uses the available capacity from a novel adaptive bandwidth estimation scheme, a congestion indicator derived from a congestion controller, Peak bit/cell Rate (PBR/PCR) estimate from new sources, along with the desired QoS metrics, and makes decisions whether to "admit" or "reject" new sources so that the QoS metrics and network utilization are met. The admission controller dynamics are expressed as a discrete-event system while the bandwidth estimator design-based on Lyapunov theory and the congestion controller dynamics are given in discrete-time. The novel aspect of the proposed approach is the application of hybrid system theory to prove the performance of the proposed admission controller, stability and the development of rigorous and repeatable design procedure. The performance of the proposed AC is evaluated using the QoS metrics, which are given in terms of service delay, packet/cell losses, and network utilization. Simulation results are presented by streaming ON/OFF and MPEG video data into the network. Results show that the proposed AC admits significantly more traffic compared to other available admission control schemes thereby guaranteeing high network utilization while maintaining the desired QoS.