We have observed the decay K-L-->mu (+)mu (-)gamma gamma at the KTeV experi
ment at Fermilab. This decay presents a formidable background to the search
for new physics in K-L-->pi (0)mu (+)mu (-). The 1997 data yielded a sampl
e of 4 signal events, with an expected background of 0.155+/-0.081 events.
The branching ratio is beta (K-L -->mu (+)mu (-)gamma gamma)= [10.4(-5.9)(7.5) (stat)+/- 0.7 (syst)] x 10(-9) with m(gamma gamma)greater than or equa
l to1 MeV/c(2), consistent with a QED calculation which predicts (9.1+/-0.8
) x 10(-9).