We present outburst spectroscopy of GX 339-4 that may reveal the motion of its elusive companion star. N III lines exhibit sharp emission components moving over ~300 km s-1 in a single night. The most plausible interpretation of these components is that they are formed by irradiation of the companion star and the velocities indicate its orbital motion. We also detect motion of the wings of the He II 4686 Å line and changes in its morphology. No previously proposed period is consistent with periodic behavior of all of these measures. However, consistent and sensible solutions are obtained for periods around 1.7 days. For the best period, 1.7557 days, we estimate a mass function of 5.8+/-0.5 Msolar. Even allowing for aliases, the 95% confidence lower limit on the mass function is 2.0 Msolar. GX 339-4 can therefore be added to the list of dynamical black hole candidates. This is supported by the small motion in the wings of the He II line; if the compact object velocity is not larger than the observed motion, then the mass ratio is q<~0.08, similar to other systems harboring black holes. Finally, we note that the sharp components are not always present but do seem to occur within a repeating phase range. This appears to migrate between our epochs of observation and may indicate shielding of the companion star by a variable accretion geometry such as a warp.