Call Admission Control (CAC) has been accepted as a potential solution for supporting diverse, heterogeneous traffic sources demanding different Quality of Services (QOSs) in ATM networks. Also, CAC is required to consume a minimum of time and space to make call acceptance decisions. In this paper, we present an efficient neutral-network-based CAC (NNCAC) mechanism for ATM networks with heterogeneous arrivals. All heterogeneous traffic calls are initially categorized into various classes. Based on the number of calls in each class, NNCAC efficiently and accurately estimates the cell delay and cell loss ratio of each class in real time by means of a pre-trained neutral network. According to our decent study which exhibits the superiority of the employment of analysis-based training data over simulation-based data, we particularly construct the training data from a heterogeneous-arrival dual-class queueing model M^[N₁] + I^[N₂]/D/1/K, where M and I represent the Bernoulli and interrupted Bernoulli processes, and N₁ and N₂ represent the corresponding numbers of calls, respectively. Analytic results of the queueing model are confirmed by simulation results. Finally, we demonstrate the profound agreement of our neural-network-based estimated results with analytic results, justifying the viability of our NNCAC mechanism.