We present experiments on the thermal transport in the spin- chain compound copper pyrazine dinitrate . The heat conductivity shows a surprisingly strong dependence on the applied magnetic field , characterized at low temperatures by two main features. The first one appearing at low is a characteristic dip located at , that may arise from umklapp scattering. The second one is a plateaulike feature in the quantum critical regime, , where is the saturation field at . The latter feature clearly points towards a momentum and field-independent mean free path of the spin excitations, contrary to theoretical expectations.