The Large Hadron Collider has extended the reach of particle-physics experiments by more than a factor of three, with a potential for discovery of new physics at the TeV energy scale. Many models of physics beyond the Standard Model predict new, massive particles with long lifetimes. Examples include supersymmetry with R-parity violation, suppressed decays of the next-to-lightest supersymmetric particle, or models with hidden sectors. The decay vertices of particles with lifetimes of order picoseconds to nanoseconds can be efficiently identified by the ATLAS detector. We present the results of a search for such particles. Based on this search, we place limits on the production cross-section of supersymmetric particles in an R-parity-violating scenario as a function of the neutralino lifetime. Limits are presented for different squark and neutralino masses, enabling extension of the limits to a variety of other models.