  (Abridged) We present the results of a new search for variable stars in the
Local Group (LG) isolated dwarf galaxy IC1613, based on 24 orbits of F475W and
F814W photometry from the ACS camera onboard the HST. 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 (RRc) pulsators, and 5 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 day, 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 first-overtone mode pulsators,
respectively. Another two are clearly second-overtone Cepheids, the first ones
to be discovered beyond the Magellanic Clouds. We estimate the distance to
IC1613 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 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.
