We have searched for gravitational waves from coalescing low mass compact binary systems with a total mass between 2M⊙and 35M⊙and a minimum component mass of 1M⊙using data from the first year of the fifth science run of the three LIGO detectors, operating at design sensitivity. Depending on the mass, we are sensitive to coalescences as far as 150 Mpc from the Earth. No gravitational-wave signals were observed above the expected background. Assuming a population of compact binary objects with a Gaussian mass distribution representing binary neutron star systems, black hole–neutron star binary systems, and binary black hole systems, we calculate the 90% confidence upper limit on the rate of coalescences to be 3.9×10^-2 yr^-1 L_10^-1, 1.1×10^-2 yr^-1 L_10^-1, and 2.5×10^-3yr^-1 L-1_10, respectively, where L_10 is 10^10times the blue solar luminosity. We also set improved upper limits on the rate of compact binary coalescences per unit blue-light luminosity, as a function of mass.