  We present new Near-Infrared (J,K) magnitudes for 114 RR Lyrae stars in the
globular cluster Omega Cen (NGC 5139) which we combine with data from the
literature to construct a sample of 180 RR Lyrae stars with J and K mean
magnitudes on a common photometric system. This is presently the largest such
sample in any stellar system. We also present updated predictions for J,K-band
Period-Luminosity relations for both fundamental and first-overtone RR Lyrae
stars, based on synthetic horizontal branch models with metal abundance ranging
from Z=0.0001 to Z=0.004. By adopting for the Omega Cen variables with measured
metal abundances an alpha-element enhancement of a factor of 3 (about 0.5 dex)
with respect to iron we find a true distance modulus of 13.70 (with a random
error of 0.06 and a systematic error of 0.06), corresponding to a distance
d=5.5 Kpc (with both random and systematic errors equal to 0.03 Kpc). Our
estimate is in excellent agreement with the distance inferred for the eclipsing
binary OGLEGC-17, but differ significantly from the recent distance estimates
based on cluster dynamics and on high amplitude Delta Scuti stars.
