The elongational viscosity of low‐molecular weight melts is not easy to measure because of the low stretching forces involved. We turn therefore to the method of gravity spinning, in which an isothermal filament is allowed to elongate under its own weight. By this means one can obtain forces of the order of tens of dynes, several orders of magnitude lower than previous work on melts. Although the experiment is easy to perform in principle (the only on‐line measurement that needs to be made is the diameter variation with length), there are difficulties in practice having to do with establishing a steady spinline with a zero end force. The nonconstant deformation along the spinline requires use of a viscoelastic model in the data analysis; therefore, two kinds of differential constitutive models were used. Three low molecular weight polypropylenes, used in the manufacture of melt‐blown fabrics, were studied at 180°C and 200°C. The basic check, that the elongational viscosity be three times the shear viscosity at low deformation rates, was achieved over a substantial part of the spinline for the thinnest of the materials. Further conclusions are not independent of the rheological model. We could not distinguish between the two differential models that were selected and chose the simpler “convected Maxwell model” for a detailed study. If a single time constant (rather than a spectrum) is used, its value needs to be larger in elongation than in shear. The time constant in elongation can be correlated in a reasonable way with molecular weight.