We present experiments on the thermal transport in the spin-1/2 chain compound copper pyrazine dinitrate Cu(C4H4N2)(NO3)2. The heat conductivity shows a surprisingly strong dependence on the applied magnetic field B, characterized at low temperatures by two main features. The first one appearing at low B is a characteristic dip located at muBB approximately kBT, that may arise from umklapp scattering. The second one is a plateaulike feature in the quantum critical regime, muB|B - Bc| < kBT, where Bc is the saturation field at T=0. The latter feature clearly points towards a momentum and field-independent mean free path of the spin excitations, contrary to theoretical expectations.