We study contention resolution in a multiple-access channel such as the Ethernet channel. In the model that we consider, n users generate messages for the channel according to a probability distribution. Raghavan and Upfal have given a protocol in which the expected "delay" (time to get serviced) of every message is O(log n) when messages are generated according to a Bernoulli distribution with generation rate up to about 1/10. We present a protocol in which the expected average message delay is O(1) when messages are generated according to a Bernoulli distribution with a generation rate smaller than 1/e. To achieve this result we first consider an analogous model in which users are synchronized (i.e., they agree about the time), there are potentially an infinite number of users, and messages are generated according to a Poisson distribution with generation rate up to 1/e. (Each message constitutes a new user.) We give a protocol in which the expected delay of any message is O(1). Next we show how to simulate the protocol using n synchronized users. Finally, we show how to build the synchronization into the protocol.