  Motivated by an amazing range of reported distances to the nearby Local Group
spiral galaxy M33, we have obtained deep near-infrared photometry for 26
long-period Cepheids in this galaxy with the ESO VLT. From the data we
constructed period-luminosity relations in the J and K bands which together
with previous optical VI photometry for the Cepheids by Macri et al. were used
to determine the true distance modulus of M33, and the mean reddening affecting
the Cepheid sample with the multiwavelength fit method developed in the
Araucaria Project. We find a true distance modulus of 24.62 for M33, with a
total uncertainty of +- 0.07 mag which is dominated by the uncertainty on the
photometric zero points in our photometry. The reddening is determined as
E(B-V)=0.19 +- 0.02, in agreement with the value used by the HST Key Project of
Freedman et al. but in some discrepancy with other recent determinations based
on blue supergiant spectroscopy and an O-type eclipsing binary which yielded
lower reddening values. Our derived M33 distance modulus is extremely
insensitive to the adopted reddening law. We show that the possible effects of
metallicity and crowding on our present distance determination are both at the
1-2% level and therefore minor contributors to the total uncertainty of our
distance result for M33.
