The clocklike properties of pulsars moving in the gravitational fields of their unseen neutron-star companions have allowed unique tests of general relativity and provided evidence for gravitational radiation. We report here the detection of the 2.8-second pulsar J0737-3039B as the companion to the 23-millisecond pulsar J0737-3039A in a highly relativistic double neutron star system, allowing unprecedented tests of fundamental gravitational physics. We observed a short eclipse of J0737-3039A by J0737-3039B and orbital modulation of the flux density and the pulse shape of J0737-3039B, probably because of the influence of J0737-3039A's energy flux on its magnetosphere. These effects will allow us to probe magneto-ionic properties of a pulsar magnetosphere.