We have constructed an apparatus which provides enhanced resolution in the measurement of the free surface profile and the axial force exiting the die during spinning of a liquid filament. In this paper we demonstrate how this information can be exploited to give quantitative information about rheological material properties in isothermal elongational flows. The fiber spinning experiments are coupled with mathematical models that serve as inverse problems to deduce material properties when the fiber profile and upstream axial force are experimentally known. We first develop integral and differential forms of the fiber spinning momentum balance which describe how stress varies down the length of the filament for an incompressible material, independent of rheology. These forms are then combined with experiments to deduce the evolution of the elongational viscosity along the spinline and verify the accuracy of free surface measurements. A fiber spinning model specialized to the Giesekus constitutive equation is then combined with spinline measurements to determine material constants within the Giesekus constitutive assumption. In particular, our technique is used to characterize the elongational rheology of a Boger test fluid. We obtain significantly different values for the relaxation time and mobility parameter than we obtain via shear rheometry; the solvent and polymer viscosities deduced from our spinline and shear experiments are essentially the same.