An experiment demonstrating the production of double- Lambda hypernuclei in (K-, K+) reactions on Be-9 was carried out at the D6 line in the BNL alternating-gradient synchrotron. The technique was the observation of pions produced in sequential mesonic weak decay, each pion associated with one unit of strangeness change. The results indicate the production of a significant number of the double hypernucleus H-4(Lambda Lambda) and the twin hypernuclei H-4(Lambda) and H-3(Lambda). The relevant decay chains are discussed and a simple model of the production mechanism is presented. An implication of this experiment is that the existence of an S = -2 dibaryon more than a few MeV below the Lambda Lambda mass is unlikely.