  We investigate the measurement of supersymmetric particle masses at the LHC
in gravitino dark matter (GDM) scenarios where the next-to-lightest
supersymmetric partner (NLSP) is the lighter scalar tau, or stau, and is stable
on the scale of a detector. Such a massive metastable charged sparticle would
have distinctive Time-of-Flight (ToF) and energy-loss ($dE/dx$) signatures. We
summarise the documented accuracies expected to be achievable with the ATLAS
detector in measurements of the stau mass and its momentum at the LHC. We then
use a fast simulation of an LHC detector to demonstrate techniques for
reconstructing the cascade decays of supersymmetric particles in GDM scenarios,
using a parameterisation of the detector response to staus, taus and jets based
on full simulation results. Supersymmetric pair-production events are selected
with high redundancy and efficiency, and many valuable measurements can be made
starting from stau tracks in the detector. We recalibrate the momenta of taus
using transverse-momentum balance, and use kinematic cuts to select
combinations of staus, taus, jets and leptons that exhibit peaks in invariant
masses that correspond to various heavier sparticle species, with errors often
comparable with the jet energy scale uncertainty.
