The problem is of interest for the creation of adequate models of ice cover dynamics in the coastal zone of the Arctic seas. Despite the simplifications adopted in the work, the proposed model allows to take into account the flow of heat from water to the front of glaciation, the dependence of the freezing temperature of sea water on its salinity, the thickness of the snow cover and the conditions on the boundaries of snow and atmosphere. A significant simplification, which allowed to obtain an exact analytical solution to the problem, is the assumption that all the salt from the formed ice at all times goes into the ice layer of water. Under the condition of smallness of the number of Stephen small and the slew rate of the ice compared to the rate of establishment of thermal equilibrium in the layer of snow, it is permissible to use quasi-stationary variant of a mathematical model. At the initial stage of ice formation, shallow water depth has little effect on the dynamics of ice growth and for shallow depths is characterized by stabilization of ice thickness, the faster, the lower the depth. For each set of physical parameters of the problem there is a limit depth, starting from which the dynamics of ice growth does not depend on the depth. The presence of a non-freezing layer of high salinity sea water is characteristic for the problems of dynamics of ice growth in shallow water.