We report the discovery of energetic (~1 MeV/nuc) H3 and H2 molecules in Jupiter's magnetosphere. The data, obtained with the LECP instrument on Voyager 2, showed these molecules to be present throughout the magnetosphere and as far as 180 RJ from the planet, in the ''magnetospheric wind'' region. Although the relative abundances of H3 and H2 do not show a monotonic trend with distance from Jupiter, the intervals of highest abundance were found in the outer magnetosphere. As an example, in the radial range 51--56 RJ of the dayside magnetosphere, the abundances of H3 and H2 (0.60--0.95 MeV/nuc) were about 20% and 13--25% that of He, respectively, and the He abundance was about 1--2% that of H. Since H+3 is expected to be an important constituent of Jupiter's ionopshere, our data provides strong evidence that, in addition to Io, the ionsphere may be an important local plasma source for the Jovian energetic particles. To our knowledge, the measurements reported here represent the first detection in nature of molecules at energies as high as 1 MeV/nucleon.