This paper presents a study of the contact properties of field effect transistors (FETs) based on single grains of the organic semiconductor sexithiophene ('6T', E-gap similar to 2.3 eV). The FETs are constructed by vapor deposition of isolated 6T grains 2-15 nm thick and 1-3 mum wide into <500 nm gaps between Au wires pre-patterned on SiO2/Si substrates. The Au wires serve as source and drain contacts to the grain and the doped Si substrate serves as the gate electrode. We show from the contact area dependence of the drain current-drain voltage characteristics that the FETs are contact limited. Both the source/6T and drain/6T interfaces can be considered to be Schottky contacts and, consequently, the device can be modeled as a pair of back-to-back diodes. Current is limited by the reverse-biased, hole injecting source/6T contact where the mechanism of hole injection from the metal to the semiconductor is best described by a drift/diffusion process with a field-dependent mobility. The contact resistance of the source/6T interface can be as high as 1 G Omega, or similar to 10(5) Omega -cm normalized for contact width, and is gate and drain voltage dependent. Greater contact resistances result when the Au electrodes are modified with self-assembled monolayers of dodecanethiol or when Cr is used instead of Au, resulting in dramatically inhibited charge transport.