We have serendipitously discovered an extremely lithium-rich star on the re
d giant branch of the globular cluster M3 (NGC 5272). An echelle spectrum o
btained with the Keck I High-Resolution Echelle Spectrograph reveals a Li I
lambda 6707 resonance doublet of 520 m Angstrom equivalent width, and our
analysis places the star among the most Li-rich giants known: log epsilon(L
i) similar or equal to +3.0. We determine the elemental abundances of this
star, IV-101, and three other cluster members of similar luminosity and col
or and conclude that IV-101 has abundance ratios typical of giants in M3 an
d M13 that have undergone significant mixing. We discuss mechanisms by whic
h a low-mass star may be so enriched in Li, focusing on the mixing of mater
ial processed by the hydrogen-burning shell just below the convective envel
ope. While such enrichment could conceivably happen only rarely, it may in
fact regularly occur during giant-branch evolution but be rarely detected b
ecause of rapid subsequent Li depletion.