  Recently a new technique for the production of muon ($\mu^+$) and muonium
($\mu^+e^-$) beams of unprecedented brightness has been proposed. As one
consequence and using a highly stable Mach-Zehnder type interferometer, a
measurement of the gravitational acceleration $\bar{g}$ of muonium atoms at the
few percent level of precision appears feasible within 100 days of running
time. The inertial mass of muonium is dominated by the mass of the positively
charged - antimatter - muon. The measurement of $\bar{g}$ would be the first
test of the gravitational interaction of antimatter, of a purely leptonic
system, and of particles of the second generation.
