  We present the first muon spin relaxation measurements ever performed on
elemental Pu, and set the most stringent upper limits to date on the magnitude
of the ordered moment in alpha-Pu and delta-stabilized Pu (alloyed with 4.3 at.
% Ga). Assuming a nominal hyperfine coupling field of 1 kOe per Bohr magneton
we set an upper limit of 0.001 Bohr magnetons for both materials at T = 4 K.
