  We use 373 hours ($\approx$ 15 days) of data from the second science run of
the LIGO gravitational-wave detectors to search for signals from binary neutron
star coalescences within a maximum distance of about 1.5 Mpc, a volume of space
which includes the Andromeda Galaxy and other galaxies of the Local Group of
galaxies. This analysis requires a signal to be found in data from detectors at
the two LIGO sites, according to a set of coincidence criteria. The background
(accidental coincidence rate) is determined from the data and is used to judge
the significance of event candidates. No inspiral gravitational wave events
were identified in our search. Using a population model which includes the
Local Group, we establish an upper limit of less than 47 inspiral events per
year per Milky Way equivalent galaxy with 90% confidence for non-spinning
binary neutron star systems with component masses between 1 and 3 $M_\odot$.
