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We report on a search for gravitational waves from coalescing compact binaries, of total mass between 2 and 35 Msun, using LIGO observations between November 14, 2006 and May 18, 2007. No gravitational-wave signals were detected. We report upper limits on the rate of compact binary coalescence as a function of total mass. The LIGO cumulative 90%-confidence rate upper limits of the binary coalescence of neutron stars, black holes and black hole-neutron star systems are 1.4x10^-2, 7.3x10^-4 and 3.6x10^-3 yr^-1L_10^-1 respectively, where L_10 is 10^10 times the blue solar luminosity.