We present the first part of a new catalog of variable stars (OIII-CVS) compiled from the data collected in the course of the third phase of the Optical Gravitational Lensing Experiment (OGLE-III). In this paper we describe the catalog of 3361 classical Cepheids detected in the ~40 square degrees area in the Large Magellanic Cloud. The sample consists of 1848 fundamental-mode (F), 1228 first-overtone (1O), 14 second-overtone (2O), 61 double-mode F/1O, 203 double-mode 1O/2O, 2 double-mode 1O/3O, and 5 triple-mode classical Cepheids. This sample is supplemented by the list of 23 ultra-low amplitude variable stars which may be Cepheids entering or exiting instability strip. The catalog data include VI high-quality photometry collected since 2001, and for some stars supplemented by the OGLE-II photometry obtained between 1997 and 2000. We provide basic parameters of the stars: coordinates, periods, mean magnitudes, amplitudes and parameters of the Fourier light curve decompositions. Individual objects of particular interest are discussed, including single-mode second-overtone Cepheids, multiperiodic pulsators with unusual period ratios or Cepheids in eclipsing binary systems. We discuss the variations of the Fourier coefficients with periods and point out on the sharp feature for periods around 0.35 days of first-overtone Cepheids, which can be explained by the occurrence of 2:1 resonance between the first and fifth overtones. Similar behavior at P=3 days for 1O Cepheids and P=10 days for F Cepheids are also interpreted as an effect of resonances between two radial modes. We fit the period-luminosity relations to our sample of Cepheids and compare these functions with previous determinations.
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