The evolution of the transient extensional stresses in dilute and semi-dilute viscoelastic polymer solutions was measured with a filament stretching rheometer of a design similar to that first introduced by Sridhar et al. The solutions were polystyrene-based Boger fluids which were stretched at constant strain rates in the range 0.6 less than or equal to epsilon(0) less than or equal to 4 s(-1) and to Hencky strains of epsilon > 4. The test fluids all strain-hardened and Trouton ratios exceeding 1000 were obtained at high strains. In addition to measuring the transient tensile stress growth, the decay of the tensile viscoelastic stress in the fluid column following cessation of uniaxial elongation was also monitored as a function of the total imposed Hencky strain and the strain rate. The measured relaxation functions were found to be significantly different from those observed following cessation of steady shear flow. The extensional stresses initially decayed very rapidly upon cessation of uniaxial elongation followed by a slower viscoelastic relaxation. For the most elastic fluids, partial decohesion of the fluid filament from the endplates of the rheometer was observed in tests conducted at high strain rates. This elastic instability is initiated near the rigid endplate fixtures of the device and it results in the progressive breakup of the fluid column into individual threads or 'fibrils' with a regular azimuthal spacing. These fibrils elongate and bifurcate as the fluid sample is elongated further. In tests conducted at the highest Deborah numbers, complete sample decohesion from the endplates and rapid elastic recoil were sometimes observed. The critical stress and strain at the onset of the instability were determined by monitoring the tensile force exerted by the filament, the sample radius, and were used to construct an approximate stability diagram. Flow visualization experiments using a modified stretching device showed that the instability develops as a consequence of an axisymmetry-breaking meniscus instability in the non-homogeneous region of highly deformed fluid near the rigid endplate.