We have collected a 43 event sample of the decay K-L --> e(+)e(-)mu (+)mu (-) with negligible backgrounds and measured its branching ratio to be (2.62 +/- 0.40 +/- 0.17) X 10(-9). We see no evidence for CP violation in this decay. In addition, we set the 90% confidence upper limit on the combined branching ratios for the lepton flavor violating decays K-L --> e(+/-)e(+/-)mu (-/+)mu (-/+) at B(K-L --> e(+/-)e(+/-)mu (-/+)mu (-/+)) less than or equal to 1.23 X 10(-10), assuming a uniform phase space distribution.