Improvements to the Weissenberg rheogoniometer are necessary in order to measure the transient rheological properties of polymer melts correctly. The improvements reported concern the mechanical design, a new heating system, a new normal force measuring system, and additional equipment for the relaxation test. Reliable short-time results require sufficiently stiff torque and normal force springs, and a small radius and relatively large angles of the cone-and-plate gap. The behavior of the LDPE melt under test is “linear viscoelastic,” if shear rate or total shear are small: The relaxation modulus, the stress growth at the onset of constant shear rate, the stress relaxation after cessation of steady shear flow, and, in addition, dynamic shear data (from an oscillation viscometer) all show consistent results when correlated by means of formulae from the theory of linear viscoelasticity. Shearing in the nonlinear range with constant shear rate leads to pronounced maxima of the shear stress p12 and of the first normal stress difference p11 − p22 which occur at constant total shear, almost independent of shear rate. Comparison of shear and tensile data (from extensional rheometer) confirms the Trouton relation in the linear-viscoelastic case. In the nonlinear case, there is a “work softening” in shear and a “work hardening” in extension.