We present the results of observations with the TAMA300 gravitational-wave detector, targeting burst signals from stellar-core collapse events. We used an excess-power filter to extract gravitational-wave candidates, and developed two methods to reduce fake events caused by non-stationary noises of the detector. These analysis methods were applied to real data from the TAMA300 interferometric gravitational wave detector. We compared the data-processed results with those of a Monte Carlo simulation with an assumed galactic-event distribution model and with burst waveforrns expected from numerical simulations of stellar-core collapses, in order to interpret the event candidates from an astronomical viewpoint. We set an upper limit of 5.0 x 10(3) events s(-1) on the burst gravitational-wave event rate in our galaxy with a confidence level of 90%.