  The extragalactic distance scale builds directly on the Cepheid
Period-Luminosity (PL) relation as delineated by the sample of Cepheids in the
Large Magellanic Cloud (LMC). However, the LMC is a dwarf irregular galaxy,
quite different from the massive spiral galaxies used for calibrating the
extragalactic distance scale. Recent investigations suggest that not only the
zero-point but also the slope of the Milky Way PL relation differ significantly
from that of the LMC, casting doubts on the universality of the Cepheid PL
relation. We want to make a differential comparison of the PL relations in the
two galaxies by delineating the PL relations using the same method, the
infrared surface brightness method (IRSB), and the same precepts. The IRSB
method is a Baade-Wesselink type method to determine individual distances to
Cepheids. We apply a newly revised calibration of the method as described in an
accompanying paper (Paper I) to 36 LMC and five SMC Cepheids and delineate new
PL relations in the V,I,J, & K bands as well as in the Wesenheit indices in the
optical and near-IR. We present 509 new and accurate radial velocity
measurements for a sample of 22 LMC Cepheids, enlarging our earlier sample of
14 stars to include 36 LMC Cepheids. The new calibration of the IRSB method is
directly tied to the recent HST parallax measurements to ten Milky Way
Cepheids, and we find a LMC barycenter distance modulus of 18.45+-0.04 (random
error only) from the 36 individual LMC Cepheid distances. We find a significant
metallicity effect on the Wvi index gamma(Wvi)=-0.23+-0.10 mag/dex as well as
an effect on the slope. The K-band PL relation on the other hand is found to be
an excellent extragalactic standard candle being metallicity insensitive in
both slope and zero-point and at the same time being reddening insensitive and
showing the least internal dispersion.
