We report Pauli spin blockade in a carbon nanotube double quantum dot defined by tunnel barriers at the contacts and a structural defect in the nanotube. We observe a pronounced current suppression for negative source-drain bias voltages, which is investigated for both symmetric and asymmetric coupling of the quantum dots to the leads. The measured differential conductance agrees well with a theoretical model of a double quantum dot system in the spin-blockade regime, which allows us to estimate the occupation probabilities of the relevant singlet and triplet states. This work shows that effective spin-to-charge conversion in nanotube quantum dots is feasible and opens the possibility of single-spin readout in a material that is not limited by hyperfine interaction with nuclear spins.