We report the results of the first search for gravitational waves from compact binary coalescence using data from the Laser Interferometer Gravitational-Wave Observatory and Virgo detectors. Five months of data were collected during the Laser Interferometer Gravitational-Wave Observatory's S5 and Virgo's VSR1 science runs. The search focused on signals from binary mergers with a total mass between 2 and 35M(circle dot). No gravitational waves are identified. The cumulative 90%-confidence upper limits on the rate of compact binary coalescence are calculated for nonspinning binary neutron stars, black hole-neutron star systems, and binary black holes to be 8: 7 X 10(-3) yr(-1) L(10)(-1), 2.2 X 10(-3) yr(-1) L(10)(-1), and 4.4 X 10(-4) yr(-1) L(10)(-1), respectively, where L(10) is 10(10) times the blue solar luminosity. These upper limits are compared with astrophysical expectations.