A layer of fluid containing gradients of both temperature and salinity is subject to several instabilities of geophysical interest. When the salinity and temperature increase upwards, the layer may become unstable even if the density profile indicates stability. This ‘doubly diffusive’ instability, first treated by Stern, is seen experimentally to consist of thin fingers of up- and downgoing fluid. Linear analysis cannot explain this small horizontal scale for a steady-state process, but a nonlinear treatment of the problem combined with a stability analysis indicates that only small-scale motions are stable when the salinity gradient is larger than that necessary for the onset of instability. In the limit of small salt diffusivity the flux of salt is calculated using the Galerkin technique and found to reach a maximum at a wavelength that decreases with increasing salinity and temperature gradients. The stability of the finite amplitude solutions is treated; only small-scale motions are found to be stable and the wavelength of the most stable mode is found to compare favourably with the wavelength that maximizes the salt flux.