The behaviour of the diffusion front of a magnetic fluid in contact with a miscible non-magnetic fluid in a normal magnetic field is studied. It was found that the magnetic field is a cause of the diffusion front bending and its movement accompanied by intense advective flows. These flows lead to the fast growth of the wavy shape of the diffusion front and formation of the peaks. This phenomenon is studied both numerically and experimentally. The reasons for the instability of the diffusion front in a magnetic field are discussed. The influence of the parameters of the problem (Schmidt number, magnetic Rayleigh number, magnetic field, the thickness of the layer, diffusion front width, etc.) on the instability parameters is studied both numerically and experimentally. It is shown that the studied instability differs from Rosensweig's instability.