The heavy-flavor-conserving nonleptonic weak decays of heavy baryons are studied in a formalism that incorporates both heavy-quark symmetry and chiral symmetry. The phenomenological S=1 nonleptonic weak chiral Lagrangian for these transitions contains two independent coupling constants that describe the transitions between two flavor-SU(3) antitriplet heavy baryons, and the transitions between two flavor-SU(3) sextet heavy baryons. In the MIT bag model and the diquark model, only transitions between antitriplets are allowed. The coupling constants for these transitions are calculated in both models. The result is applied to specific nonleptonic decays such as cc, and the branching ratios are found to be of the order of 10-4. An example of a nonleptonic decay due to symmetry breaking is provided by cc, which is estimated to have a smaller branching ratio, of the order of 10-5.