Three-dimensional direct numerical simulations are performed that give us an in-depth account of the evolution and structure of the double-diffusive interface. We examine the diffusive convection regime, which, in the oceanographically relevant case, consists of relatively cold fresh water above warm salty water. A 'double-boundary-layer' structure is found in all of the simulations, in which the temperature (T) interface has a greater thickness than the salinity (S) interface. Therefore, thin gravitationally unstable boundary layers are maintained at the edges of the diffusive interface. The TS-interface thickness ratio is found to scale with the diffusivity ratio in a consistent manner once the shear across the boundary layers is accounted for. The turbulence present in the mixed layers is not able to penetrate the stable stratification of the interface core, and the TS-fluxes through the core are given by their molecular diffusion values. Interface growth in time is found to be determined by molecular diffusion of the S-interface, in agreement with a previous theory. The stability of the boundary layers is also considered, where we find boundary layer Rayleigh numbers that are an order of magnitude lower than previously assumed.