An experiment demonstrating the production of double-Lambda hypernuclei in (K(-),K(+)) reactions on (9)Be 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 (4)(double Lambda)H and the twin hypernuclei (4)(Lambda)H and (3)(Lambda)H. 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 double Lambda mass is unlikely.