Slow growth of calcite in confinement is abundant in Nature and man made materials. There is ample evidence that such confined growth may create forces that fracture solids. The thermodynamic limits are well known but since confined crystal growth is transport limited and difficult to control in experiment we have almost no information on the mechanisms or limits of these processes. We present a novel approach to in situ study of confined crystal growth using microfluidics for accurate control of the saturation state of the fluid and interferometric measurement of the topography of the growing confined crystal surface. We observe and explain the diffusion limited confined growth structures observed and can measure the crystal "floating" on a fluid film of 10-40~nm thickness due to the disjoining pressure. We find that there are two end member behaviours: smooth or intermittent growth in the contact region, the latter being faster than the former.