We present algorithms for connections admission control (CAC) in an ATM network based upon the use of on-line measurement and estimation. (The algorithms described in this paper are the subject of a UK patent application.) Our approach uses the theory of large deviations, which is a valuable tool for reasoning about rare events in stochastic systems. We have shown previously that the large deviations rate function, or entropy, of ATM traffic can be estimated from activity measurements. This can be used to determine the effective bandwidth of the traffic. In this paper we present practical measurement-based CAC algorithms which are computationally efficient, and can be used with traffic whose statistical characteristics are unknown.