The flow curves and kinematics of stretch-thickening fluids flowing through an orifice die have been determined under creeping-flow conditions. As shown, the advantage of this particular flow geometry over other contraction flows is that the pressure drop P(g) across the orifice is related to energy losses in such a way that experimental data can be readily analysed in terms of pronounced elongational effects. The fluids tested were low concentration solutions of flexible coils (salt-containing solutions of partially hydrolysed polyacrylamide (HPAM) and polyethylene oxide (PEO) solutions) and of semi-rigid rods (unsalted HPAM solutions) in a thick solvent. Flow parameters were varied over the whole range, that is from Newtonian behaviour at low flow conditions to very strong flow conditions capable of producing chain breakup. Three flow regimes, interconnected by smooth transitions, were identified for solutions of flexible coils. The feature of each regime is that P(g) varies with flow rate q(v) independently of polymer concentration. The flow structure over each regime was thoroughly documented. Careful simultaneous visualisations showed that small-scale instabilities do exist under conditions where the upstream flow may seem to possess a mean stable axisymmetric structure. Stress growth in each regime was connected to a distinct structural mechanism. In the initial linear-viscous Newtonian regime, where P(g) is linear in q(v), chains are deformed slightly and solvent viscous stresses prevail. In the intermediate quadratic regime, where P(g) varies with q(v)2, significant chain unravelling takes place, giving rise to high elastic stresses. In the ultimate linear-viscous regime where P(g) is again linear in q(v), chains are stretched to their full length and viscous stresses due to hydrodynamic interaction arc dominant. The quadratic regime is absent for solutions of semi-rigid rods. However, these solutions show the same quantitative behaviour in the ultimate regime as solutions of flexible coils, indicating that similar structural effects govern stress growth in this regime. This work gives a global view of low concentration polymer solution flow through an orifice die. Moreover, it provides a basis for incorporating realistic structural effects in constitutive models.