  We report on a search for gravitational waves from coalescing compact binary
systems in the Milky Way and the Magellanic Clouds. The analysis uses data
taken by two of the three LIGO interferometers during the first LIGO science
run and illustrates a method of setting upper limits on inspiral event rates
using interferometer data. The analysis pipeline is described with particular
attention to data selection and coincidence between the two interferometers. We
establish an observational upper limit of $\mathcal{R}<$1.7 \times 10^{2}$ per
year per Milky Way Equivalent Galaxy (MWEG), with 90% confidence, on the
coalescence rate of binary systems in which each component has a mass in the
range 1--3 $M_\odot$.
