We report the results of high-field, low-temperature mu SR measurements of the quasi two-dimensional organic superconductors kappa -(ET)2 Cu(NCS)2 and kappa -(ET)2Cu[N(CN)2]Br. The mu SR lineshapes for these compounds indicate the existence of partially-ordered vortex lattice phases in the high magnetic field regime, up to 2.5 T for the former compound and 4 T for the latter compound. The observed sharp loss of order is found to be consistent with a vortex-lattice melting transition that is predicted by numerical simulations of weakly coupled layers of pancake vortices. It is argued that the robustness of the partially-ordered vortex lattice phases could be due to strong flux-line pinning by a dilute ensemble of defects.