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We present results of an extensive photometric search for planetary and low-luminosity object transits in the Galactic disk stars commencing the third phase of the Optical Gravitational Lensing Experiment -- OGLE-III. Photometric observations of three fields in the direction of the Galactic center (800 epochs per field) were collected on 32 nights during time interval of 45 days. Out of the total of 5 million stars monitored, about 52~000 Galactic disk stars with photometry better than 1.5% were analyzed for flat-bottomed eclipses with the depth smaller than 0.08 mag. Altogether 46 stars with transiting low-luminosity objects were detected. For 42 of them multiple transits were observed, a total of 185, allowing orbital period determination. Transits in two objects: OGLE-TR-40 and OGLE-TR-10, with the radii ratio of about 0.14 and estimate of the radius of the companion 1.0-1.5R_Jup, resemble the well known planetary transit in HD 209458. The sample was selected by the presence of apparent transits only, with no knowledge on any other properties. Hence, it is very well suited for general study of low-luminosity objects. The transiting objects may be Jupiters, brown dwarfs, or M dwarfs. Future determination of the amplitude of radial velocity changes will establish their masses, and will confirm or refute the reality of the so called ``brown dwarf desert''. The low-mass stellar companions will provide new data needed for the poorly known mass-radius relation for the lower main sequence. All photometric data are available to the astronomical community from the OGLE Internet archive.