We analyze 25 hours of data from the LIGO 40-meter prototype laser interferometric gravitational wave detector taken during the third week of November 1994. The instrument was sensitive enough to detect the gravitational-wave chirps that would be emitted by coalescing compact binary systems within our Galaxy. The data stream was searched using optimal matched filtering - the first implementation/test of this technique on real interferometric data. These methods will be an important part of upcoming LIGO data analysis. An upper limit on the rate R of neutron star binary inspirals in our Galaxy is obtained: with 90% confidence, R< 0.5/hour. Similar experiments with LIGO interferometers will provide constraints on the population of tight binary neutron star systems in the Universe.