We present the results of a new search for variable stars in the Local Group (LG) isolated dwarf galaxy IC 1613, based on 24 orbits of F475W and F814W photometry from the Advanced Camera for Surveys (ACS) on board the Hubble Space Telescope. We detected 259 candidate variables in this field, of which only 13 (all of them bright Cepheids) were previously known. Out of the confirmed variables, we found 90 RR Lyrae stars, 49 classical Cepheids (including 36 new discoveries), and 38 eclipsing binary stars for which we could determine a period. The RR Lyrae include 61 fundamental (RRab) and 24 first-overtone (FO, RRc) pulsators, and five pulsating in both modes simultaneously (RRd). As for the majority of LG dwarfs, the mean periods of the RRab and RRc (0.611 and 0.334 days, respectively) as well as the fraction of overtone pulsators (f c = 0.28) place this galaxy in the intermediate regime between the Oosterhoff types. From their position on the period-luminosity diagram and light-curve morphology, we can unambiguously classify 25 and 14 Cepheids as fundamental and FO mode pulsators, respectively. Another two are clearly second-overtone Cepheids, the first ones to be discovered beyond the Magellanic Clouds. Among the remaining candidate variables, five were classified as δ-Scuti and five as long-period variables. Most of the others are located on the main sequence, the majority of them likely eclipsing binary systems, although some present variations similar to pulsating stars. We estimate the distance to IC 1613 using various methods based on the photometric and pulsational properties of the Cepheids and RR Lyrae stars. The values we find are in very good agreement with each other and with previous estimates based on independent methods. When corrected to a common reddening of E(B - V) = 0.025 and true Large Magellanic Cloud (LMC) distance modulus of (m-M)LMC,0 = 18.515 ± 0.085, we find that all the distance determinations from the literature converge to a common value of (m-M)0 = 24.400 ± 0.014 (statistical) or 760 kpc. The parallel WFPC2 field, which lies within three core radii, was also searched for variable stars. We discovered nine RR Lyrae stars (four RRab, four RRc, and one RRd) and two Cepheids, even though the lower signal-to-noise ratio of the observations did not allow us to measure their periods as accurately as for the variables in the ACS field of view. We provide their coordinates and approximate properties for completeness. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program 10505.