A new rheometer was developed which can measure melt viscosity at constant strain rate elongation with superimposed sinusoidal oscillations (in short, superimposed elongations). The superimposed elongations were controlled by a pulse motor and a personal computer which drove rotating clamps in the rheometer. High density polyethylene (HDPE) with a bimodal molecular weight distribution and low density polyethylene (LDPE) melts were studied to assess structural changes in elongation. Linearity of stress response to the superimposed strain rate was confirmed below elongational strains of 2.8 for LDPE melt, and 2.0 for HDPE melt, when the constant strain rate was set to 0.08 s−1and the amplitude of the superimposed strain rate was set to below 0.02 s−1. Dynamic elongational moduli (E’,E‘) of HDPE and LDPE melts were calculated from stress response at the superimposed elongations by analogy to dynamic shear deformations. Strain dependence of dynamic elongational moduli of HDPE melt and LDPE melt showed steep increases at the inception of the nonlinear viscoelastic region. The dynamic elongational moduli were estimated in the linear viscoelastic region and they were found to be larger than the corresponding dynamic shear moduli.