The reproductive ratio, R0, is a fundamental quantity in epidemiology, which determines the initial increase in an infectious disease in a susceptible host population. In most epidemic models, there is a specific value of R0, the epidemic threshold, above which epidemics are possible, but below which epidemics cannot occur. As the complexity of an epidemic model increases, so too does the difficulty of calculating epidemic thresholds. Here we derive the reproductive ratio and epidemic thresholds for susceptible-infected-recovered (SIR) epidemics in a simple class of dynamic random networks. As in most epidemiological models, R0 depends on two basic epidemic parameters, the transmission and recovery rates. We find that R0 also depends on social parameters, namely the degree distribution that describes heterogeneity in the numbers of concurrent contacts and the mixing parameter that gives the rate at which contacts are initiated and terminated. We show that social mixing fundamentally changes the epidemiological landscape and, consequently, that static network approximations of dynamic networks can be inadequate.