Observations are presented which show that Li lambda 6708 is detected with an equivalent width of 0.53 +/- 0.05 Angstrom in the proper-motion object LP 944-20 (which is also known as BRI0337-3535). Hoc is detected in emission at an equivalent width of 1.2 +/- 0.5 Angstrom. The detection of Li implies a mass less than 0.065 M., making this object a brown dwarf. Moreover, the relative weakness of this Li detection (compared to the equivalent widths of 1-2 Angstrom seen in objects of similar spectral type in the Pleiades) implies that Li has been somewhat depleted. This, together with the precisely determined luminosity of LP944-20, implies a mass between 0.056 and 0.064 M., and age between 475 and 650 Myr. This makes it the first brown dwarf to have a well-constrained mass and age determined.