  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.
