We present results of the search for Cepheids in the galaxy IC1613 carried out as a sub-project of the OGLE-II microlensing survey. 138 Cepheids were found in the 14.2x14.2 arcmin region in the center of the galaxy. We present light curves, VI photometry and basic data for all these objects, as well as color-magnitude diagram of the observed field. The Period--Luminosity (PL) diagrams for IC1613 fundamental mode Cepheids for VI and interstellar extinction insensitive index W_I are constructed. Comparison of PL relations in metal poor galaxy IC1613 ([Fe/H]~-1.0 dex) with relations in metal richer Magellanic Clouds allows us to study dependence of Cepheid PL relations on metallicity in the wide range of metallicities covered by these three galaxies. The slopes of PL relations in IC1613 are identical as in the Magellanic Clouds. The comparison of brightness of Cepheids with the magnitudes of the tip of the red giant branch stars and RR Lyr stars in all three objects provides information on the stability of zero points of PL relations in the I and V-band, respectively. We find that the zero points of PL relations are constant to better than $pm0.03$ mag over the entire range of covered metallicities. Thus, the most important conclusion of the paper is that the Cepheid PL relations do not depend on metallicity. Additionally we determine the mean distance to IC1613, based on the common distance scale resulting from four major stellar distance indicators: Cepheids, RR Lyr, TRGB and red clump stars. The distance modulus to IC1613 is equal to (m-M)_IC1613=24.20 mag with the standard deviation of +/-0.02 mag from six measurements and systematic uncertainty of calibrations of +/-0.07 mag.
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