The double pulsar system, PSR J0737-3039A/B, is unique in that both neutron stars are detectable as radio pulsars. This, combined with significantly higher mean orbital velocities and accelerations when compared to other binary pulsars, suggested that the system would become the best available testbed for general relativity and alternative theories of gravity in the strong-field regime. 1Here we report on precision timing observations taken over the 2.5 years since its discovery and present four independent strong-field tests of general relativity. Use of the theory-independent mass ratio of the two stars makes these tests uniquely different from earlier studies. By measuring relativistic corrections to the Keplerian discription of the orbital motion, we find that the "postKeplerian" parameter s agrees with the value predicted by Einstein's theory of general relativity within an uncertainty of 0.05%, the most precise test yet obtained. We also show that the transverse velocity of the system's center of mass is extremely small. Combined with the system's location near the Sun, this result suggests that future tests of gravitational theories with the double pulsar will supersede the best current Solar-system tests. It also implies that the second-born pulsar may have formed differently to the usually assumed core-collapse of a helium star.