We describe a theoretical and experimental study of an axisymmetric viscous gravity current with a constant flux, confined to the space between two horizontal parallel plates. The effect of confinement results in two regions of flow: an inner region where the fluid is in contact with both plates and an outer annular region where the fluid forms a gravity current along the lower plate. We present a simple theoretical model that describes the flow dynamics by a single dimensionless parameter $J$, which is the ratio of the characteristic height of an unconfined gravity current to the height of the confined space. Theoretical height profiles display the same characteristics as unconfined gravity currents until $J \approx 0.48$, where a rapid change in behaviour occurs as confinement comes into effect. For larger values of $J$, the confined viscous gravity current gradually tends to Hele-Shaw flow, with the transition essentially complete by $J \approx 2$. We compare the findings from our theoretical model with the results of a series of experiments using golden syrup with various fluxes and gap spacings. Although the data aligns with the major aspects of the model, it is clear that other physics is at play and a single non-dimensional parameter is not sufficient to capture the flow behaviour fully. We speculate on the factors absent in our model that may be responsible for this mismatch.