Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size n. A Markovian SIR (susceptible -> infective -> recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where n -> infinity, keeping the basic reproduction number R-0 as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than 1/log n. It is shown that, as n -> infinity, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process S = {S(t); t >= 0} describing the limiting fraction of the population that are susceptible. The process S grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the start of the regenerative cycle. Properties of the process S, including the jump size and stationary distributions, are determined.