The interfacial thickness and mechanical strength of several model interfaces between immiscible polymer melts were investigated. The interfacial width at equilibrium was determined by neutron reflectivity, and the adhesion energy was determined with an asymmetric probe test method inspired of the tests used for soft adhesives. A very good correlation was found between the adhesion energy and the measured interfacial width in a qualitatively similar way as what was observed for interfaces between glassy polymers. For polymer pairs with chi values ranging from 0.002 to 0.05, the mechanical strength of the interface was controlled by the degree of interpenetration, or in other terms, by the interfacial free energy between the two polymers. For lower values of chi, the interfacial strength was comparable to the fracture strength of the weakest of the two bulk polymers, while for higher values of chi, the thermodynamic work of adhesion and the polymer surface mobility are probably controlling the interfacial strength.