This paper proposes an auto regressive moving average (ARMAX)-based adaptive control methodology to prevent congestion in high-speed asynchronous transfer mode (ATM) networks. An adaptive controller is developed to control traffic where sources adjust their transmission rates in response to the feedback information from the network switches. Specifically, the buffer dynamics at a given switch is modeled as a nonlinear discrete-time system and an ARMAX controller is designed so as to predict the explicit values of the transmission rates of the sources so as to prevent congestion. Tuning methods are provided for the unknown coefficients of the ARMAX model to estimate the unpredictable and statistically fluctuating network traffic. Mathematical analysis is given to demonstrate the stability of the closed-loop system so that a desired quality of service (QoS) can be guaranteed. The QoS is defined in terms of cell loss ratio (CLR), transmission delay and buffer utilization. We derive design rules mathematically for selecting the parameters of the ARMAX algorithm such that the desired performance is guaranteed during congestion and potential tradeoffs are shown. Simulation results are provided to justify the theoretical conclusions for multiple source/single switch scenarios using both ON/OFF and MPEG data. The performance of the proposed congestion control scheme is also evaluated in the presence of feedback delays for robustness considerations.