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. Our sample of Cepheids is cross-identified with previously published catalogs of these variables in the LMC. 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
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