We present studies on the thermalisation of H3+ ions in a cold He/Ar/H2 plasma at temperatures 30-70 K. We show that we are able to generate a rotationally thermalised H3+ ensemble with a population of rotational and nuclear spin states corresponding to a particular ion translational temperature. By varying the para-H2 fraction used in the experiment we are able to produce para-H3+ ions with fractional populations higher than those corresponding to thermodynamic values. At 35 K, only the lowest rotational states of para and ortho H3+ are populated. This is the first step towards experimental studies of electron-molecular ion recombination processes with precisely specified quantum states at astrophysically relevant temperatures.