One of the central predictions of metric theories of gravity, such as general relativity, is that a clock in a gravitational potential U will run more slowly by a factor of 1 + U/c(2), where c is the velocity of light, as compared to a similar clock outside the potential. This effect, known as gravitational redshift, is important to the operation of the global positioning system, timekeeping and future experiments with ultra-precise, space-based clocks (such as searches for variations in fundamental constants). The gravitational redshift has been measured using clocks on a tower, an aircraft and a rocket, currently reaching an accuracy of 7 x 10(-5). Here we show that laboratory experiments based on quantum interference of atoms enable a much more precise measurement, yielding an accuracy of 7 x 10(-9). Our result supports the view that gravity is a manifestation of space-time curvature, an underlying principle of general relativity that has come under scrutiny in connection with the search for a theory of quantum gravity. Improving the redshift measurement is particularly important because this test has been the least accurate among the experiments that are required to support curved space-time theories.