A hydrodynamic stability analysis was carried out to better understand the phenomenon of interfacial instability in the stratified flow of two viscoelastic fluids through a rectangular duct, using the Coleman‐Noll second‐order fluid as a constitutive equation. Because of the complexity of the equations to be dealt with, we considered, instead of a full three‐dimensional treatment of the disturbance, the two sets of two‐dimensional disturbances, namely (a) disturbances spatially dependent only upon thex1andx2coordinates, and (b) disturbances spatially dependent only upon thex2andx3coordinates. For each set of disturbances, approximate solutions of the governing system differential equations were obtained by using the perturbation technique. It has been found that the occurrence of interfacial instability in thex1−x2plane depends only on the viscosity ratio of the two fluids (i.e., independent of the fluid elasticity), whereas the occurrence of interfacial instability in thex1−x2plane depends on both the viscosity and elasticity ratios of the two fluids concerned. Regions of stability were constructed in terms of the viscosity and elasticity ratios of the two fluids. The stability regions determined may be used for guidance in judiciously choosing the viscosity and elasticity ratios, and the composition ratio of the two fluids flowing side by side in order to avoid an irregular interface.