THE strong form of Einstein's equivalence principle, which dictates that gravitational binding energy will suffer the same acceleration in a uniform gravitational field as all other forms of matter and energy, can be tested by comparing the accelerations of the Earth and the Moon towards the Sun. If the Earth's gravitational binding energy, which contributes ~5 parts in 1010 of its total mass, does not gravitate in the same way as other kinds of mass and energy, the Moon's orbit will be distorted in a characteristic way with an amplitude large enough to be detected by lunar laser-ranging data. But as Nordtvedt1 has pointed out, such a test is accurate only to the extent that no interfering force, such as a very feeble composition-dependent 'fifth force', fortuitously cancels a gravitational binding energy anomaly. Here we review laboratory upper limits on 'fifth forces' to set an upper limit of 1 part in 1012 (95% confidence level) on the Earth-Moon acceleration difference due to non-gravitational forces. Existing lunar-ranging data then verify the strong equivalence principle to an accuracy of 3%, and improved analyses of the lunar data should bring this precision down to the 0.2% level.