The present invention is directed to an asphalt paving system comprising an asphalt paving machine, a plurality of dispensing shafts, and a roll of pavement reinforcing membrane. The asphalt paving machine comprising a hopper, a screed, and support tracks. The plurality of dispensing shafts are mounted on the asphalt paving machine. Each roll of pavement reinforcing membrane is supported by at least one of the plurality of dispensing shafts. The plurality of mounted membrane dispensing shafts dispense the pavement reinforcing membrane continuously beneath the asphalt paving machine as the asphalt paving moves forward.