Conjugated polymers and small organic molecules are enabling new, flexible, large-area, low-cost optoelectronic devices, such as organic light-emitting diodes, transistors and solar cells. Owing to their exceptionally long spin lifetimes, these carbon-based materials could also have an important impact on spintronics, where carrier spins play a key role in transmitting, processing and storing information. However, to exploit this potential, a method for direct conversion of spin information into an electric signal is indispensable. Here we show that a pure spin current can be produced in a solution-processed conducting polymer by pumping spins through a ferromagnetic resonance in an adjacent magnetic insulator, and that this generates an electric voltage across the polymer film. We demonstrate that the experimental characteristics of the generated voltage are consistent with it being generated through an inverse spin Hall effect in the conducting polymer. In contrast with inorganic materials, the conducting polymer exhibits coexistence of high spin-current to charge-current conversion efficiency and long spin lifetimes. Our discovery opens a route for a new generation of molecular-structure-engineered spintronic devices, which could lead to important advances in plastic spintronics.