We have measured accurate near-infrared magnitudes in the J and K bands of 39 Cepheid variables in IC 1613 with well-determined periods and optical VI light curves. Using the template light curve approach of Soszy{n}ski, Gieren and Pietrzy{n}ski, accurate mean magnitudes were obtained from these data which allowed to determine the distance to IC 1613 relative to the LMC from a multiwavelength period-luminosity solution in the optical VI and near-IR JK bands, with an unprecedented accuracy. Our result for the IC 1613 distance is $(m-M)_{0} = 24.291 pm 0.014$ (random error) mag, with an additional systematic uncertainty smaller than 2%. From our multiwavelength approach, we find for the total (average) reddening to the IC 1613 Cepheids $E(B-V) = 0.090 pm 0.007$ mag,which is significantly higher than the foreground reddening of about 0.03 mag,showing the presence of appreciable dust extinction inside the galaxy. Our data suggest that the extinction law in IC 1613 is very similar to the galactic one.Our distance result agrees, within the uncertainties, with two earlier infrared Cepheid studies in this galaxy of Macri et al. (from HST data on 4 Cepheids), and McAlary et al. (from ground-based H-band photometry of 10 Cepheids), but our result has reduced the total uncertainty on the distance to IC 1613 (relative to the LMC) to less than 3%. With distances to nearby galaxies from Cepheid infrared photometry at this level of accuracy, which are currently being obtained in our Araucaria Project, it seems possible to significantly reduce the systematic uncertainty of the Hubble constant as derived from the HST Key Project approach, by improving the calibration of the metallicity effect on PL relation zero points, and by improving the distance determination to the LMC.
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