The Atlantic Meridional Overturning Circulation (AMOC) is contributing considerably to the mild climate of north-western Europe by transporting heat from the low latitudes northward. On long timescales, the AMOC is limited by the amount of potential energy that is put into the ocean by vertical mixing. By assuming a constant rate of mixing, most state-of-the-art climate models do not represent the effect of vertical mixing well, since theoretical arguments and measurements indicate that the vertical mixing strongly depends on local physical parameters - above all, on stratification. We implemented stratification-dependent mixing into a coupled climate model, and found the sensitivity of the AMOC to freshwater forcing to depend critically on the coupling between stratification and mixing. Weak coupling reproduces results from previous studies that assume constant vertical mixing. Stronger (and more realistic) coupling however allows for a positive feedback in the high northern latitudes which creates a strongly stratified layer. This layer is not penetrated by winter deep convection. Thus, the overturning north of the Greenland-Scotland Ridge is stopped.