In spintronics, pure spin currents play a key role in transmitting, processing and storing information. A pure spin current is a flow of electron spin angular momentum without a simultaneous flow of charge current. It can be carried by conduction electrons or magnons and has been studied in many inorganic metals, semiconductors and insulators, but not yet in organic semiconductors. Charge carriers in π-conjugated organic materials are localized spin-1/2 polarons which move by hopping, but the mechanisms of their spin transport and relaxation are not well understood. Here we use ferromagnetic resonance spin pumping in a ferromagnet/conjugated polymer/nonmagnetic spin-sink trilayer to demonstrate the ability of polarons to carry pure spin currents over hundreds of nanometres with long spin relaxation times of up to a millisecond and to exhibit Hanle precession. By systematically comparing charge and spin transport on the same trilayer we show that spin-orbit coupling mediates spin relaxation at room temperature.