Using a 281 pb{sup -1} data sample collected at the {psi}(3770) with the CLEO-c detector, we present the first absolute branching fraction measurement of the decay D{sup 0}{yields}K{sup -}{pi}{sup +}{pi}{sup -}e{sup +}{nu}{sub e} at a statistical significance of about 4.0 standard deviations. We find 10 candidates consistent with the decay D{sup 0}{yields}K{sup -}{pi}{sup +}{pi}{sup -}e{sup +}{nu}{sub e}. The probability that a background fluctuation accounts for this signal is less than 4.1x10{sup -5}. We find B(D{sup 0}{yields}K{sup -}{pi}{sup +}{pi}{sup -}e{sup +}{nu}{sub e})=2.8{sub -1.1}{sup +1.4}(stat){+-}0.3(syst)x10 = {sup -4}. By restricting the invariant mass of the hadronic system to be consistent with K{sub 1}(1270), we obtain the product of branching fractions B(D{sup 0}{yields}K{sub 1}{sup -}(1270)e{sup +}{nu}{sub e})xB(K{sub 1}{sup -}(1270){yields}K{sup -}{pi}{sup +}{pi}{sup -})=2.5{sub -1.0} = {sup +1.3}(stat){+-}0.2(syst)x10{sup -4}. Using B(K{sub 1}{sup -}(1270){yields}K{sup -}{pi}{sup +}{pi}{sup -})=(33{+-}3)%, we obtain B(D{sup 0}{yields}K{sub 1}{sup -}(1270)e{sup +}{nu}{sub e})=7.6{sub -3.0}{sup +4.1}(stat){+-}0.6(syst){+-}0.7 = x10{sup -4}. The last error accounts for the uncertainties in the measured K{sub 1}{sup -}(1270){yields}K{sup -}{pi}{sup +}{pi}{sup -} branching fractions.