A thin dark thread is observed in a UV/EUV solar jet in the 171A, 193A and 211A and partially in 304A. The dark thread appears to originate in the chromosphere but its temperature does not appear to lie within the passbands of the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO). We therefore implement solar magnetoseismology to estimate the plasma parameters of the dark thread. A propagating fast kink (transverse) wave is observed to travel along the dark thread. The wave is tracked over a range of ~ 7000 km by placing multiple slits along the axis of the dark thread. The phase speed and amplitude of the wave are estimated and magnetoseismological theory is employed to determine the plasma parameters. We are able to estimate the plasma temperature, density gradient, magnetic field gradient and sub-resolution expansion of the dark thread. The dark thread is found to be cool, T ~ 3x10^4 K with both strong density and magnetic field gradients. The expansion of the flux tube along its length is ~ 300-400 km.