Neutrinoless double beta decay (0 nu beta beta) induced by superparticle exchange is investigated. Such a supersymmetric (SUSY) mechanism of 0 nu beta beta decay arises within SUSY theories with R-parity nonconservation (R(p)). We consider the minimal supersymmetric standard model (MSSM) with explicit R(p) terms in the superpotential (R(p) MSSM). The decay rate for the SUSY mechanism of 0 nu beta beta decay is calculated. Numerical values for nuclear matrix elements for the experimentally most interesting isotopes are calculated within the proton-neutron quasiparticle random phase approximation. Constraints on the R(p) MSSM parameter space are extracted from current experimental half-life limits. The most stringent limits are derived from data on Ge-76. It is shown that these constraints are more stringent than those from other low-energy processes and are competitive with or even more stringent than constraints expected from accelerator searches.