A design is presented of a laboratory experiment that could test the suitability of relativistic gravity for propulsion of spacecraft to relativistic speeds. An exact time-dependent solution of Einstein's gravitational field equation confirms that even the weak field of a mass moving at relativistic speeds could serve as a driver to accelerate a much lighter payload from rest to a good fraction of the speed of light. The time-dependent field of ultrarelativistic particles in a collider ring is calculated. An experiment is proposed as the first test of the predictions of general relativity in the ultrarelativistic limit by measuring the repulsive gravitational field of bunches of protons in the Large Hadron Collider (LHC). The estimated 'antigravity beam' signal strength at a resonant detector of each proton bunch is 3 nm/s^2 for 2 ns during each revolution of the LHC. This experiment can be performed off-line, without interfering with the normal operations of the LHC.