We consider families of one-dimensional maps on the circle which mix at sub-polynomial rates. Such maps will have an indifferent fixed point, and we show that the rate of mixing of these maps is determined by the precise degeneracy of the fixed point. By constructing suitable induced maps with a countable Markov partition and applying a probabilistic coupling argument we are able to give good estimates on the rate of mixing for these families of maps. The underlying question we pose is the following: given any specific rate of mixing can we construct a map which has that rate of mixing? For a large class of specific rates we explain how to construct the appropriate map.