We report on a search for gravitational waves from coalescing compact binaries, of total mass between 2 and 35M⊙, 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 1.4×10^-2, 7.3×10^-4 and 3.6×10^-3 yr^-1 L_10^-1, respectively, where L_10 is 10^10 times the blue solar luminosity.