The nonmesonic weak decay of double-hypernuclei is studied within a microscopic diagrammatic approach. In addition to the nucleon-induced mechanism, N → nN and N N → nN N , widely studied in singlehypernuclei, additional hyperon-induced mechanisms, → n, → 0 n, and → − p, are accessible in double-hypernuclei and are investigated here. As in previous works on single-hypernuclei, we adopt a nuclear matter formalism extended to finite nuclei via the local density approximation and a one-meson exchange weak transition potential (including the ground-state pseudoscalar and vector octets mesons) supplemented by correlated and uncorrelated two-pion-exchange contributions. The weak decay rates are evaluated for hypernuclei in the region of the experimentally accessible light hypernuclei 10 Be and 13 B. Our predictions are compared with a few previous evaluations. The rate for the → n decay is dominated by K-, K * -, and η-exchange and turns out to be about 2.5% of the free decay rate, free , while the total rate for the → 0 n and → − p decays, dominated by π -exchange, amounts to about 0.25% of free . The experimental measurement of these decays would be essential for the beginning of a systematic study of the nonmesonic decay of strangeness −2 hypernuclei. This field of research could also shed light on the possible existence and nature of the H dibaryon.