An approach directed at rapid sequestration and disposal of technetium-99 from UREX (uranium extraction) liquid waste streams is presented. This stream is generated during reprocessing of light-water-reactor spent fuel to recycle the actinides and separate fission products for waste disposal. U and Tc are co-extracted from a nitric acid solution using tri-n-butylphosphate in dodecane, so that Tc(VII) is present in the strip solution after the actinide separations. The goal is to separate uranyl from the pertechnetate in this U-Tc stream and then sequester Tc in the metallic form. Our approach is based on reductive stripping of pertechnetate either from aqueous solution (for column extractions) or organic solvents (for liquid-liquid extractions). In both of these methods, metallic zinc in the presence of formic acid serves as a reducing agent, and 99Tc is recovered as a co-precipitate of Zn(II) hydroxide and hydrous Tc(IV) oxide, with a Zn:Tc ratio between 1:1 and 2:1mol/mol. This solid residue can be reduced to a Zn-Tc alloy by high temperature (500-700 degrees C) hydrogenation, and the resulting heterophase alloy can be added to a metallic Fe-Zr-Mo waste form that is processed at 1600 degrees C, with subsequent loss of Zn by evaporation. Alternatively, Zn and Tc can be separated and 99Tc sequestered as NH4TcO4 for further reduction to Tc(0) metal. The aqueous Zn reduction process removes 90% of 99Tc per cycle. The nonaqueous Zn reduction in 1:1 methanol - formic acid removes 60-70% of 99Tc per cycle, depending on the extracting agent (such as a tetraalkylammonium nitrate). The extracting agent is recycled in the process. The pertechnetate is extracted from the aqueous phase into 1,2-dichloroethane, which is removed by evaporation and reused. The residue is either calcined and steam reformed to Tc(0) or processed by the nonaqueous Zn reduction method. These methods can be used not only to remove the pertechnetate from the U-Tc product stream, but also to sequester the pertechnetate from aqueous waste streams generated through the processes described in this paper, thereby closing the cycle. The same approaches can be used to close the 99Tc cycle for other methods that are currently being developed at Los Alamos and Argonne National Laboratories.