  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. Here 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 description of the orbital motion, we
find that the ``post-Keplerian'' 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.
