  Classical Cepheids provide the foundation for the empirical extragalactic
distance ladder. Milky Way Cepheids are the only stars in this class accessible
to trigonometric parallax measurements. However, the parallaxes of Cepheids
from the second Gaia data release (GDR2) are affected by systematics because of
the absence of chromaticity correction, and occasionally by saturation. As a
proxy for the parallaxes of 36 Galactic Cepheids, we adopt either the GDR2
parallaxes of their spatially resolved companions or the GDR2 parallax of their
host open cluster. This novel approach allows us to bypass the systematics on
the GDR2 Cepheids parallaxes that is induced by saturation and variability. We
adopt a GDR2 parallax zero-point (ZP) of -0.046 mas with an uncertainty of
0.015 mas that covers most of the recent estimates. We present new Galactic
calibrations of the Leavitt law in the V, J, H, K_S , and Wesenheit W_H bands.
We compare our results with previous calibrations based on non-Gaia
measurements and compute a revised value for the Hubble constant anchored to
Milky Way Cepheids. From an initial Hubble constant of 76.18 +/- 2.37 km/s/Mpc
based on parallax measurements without Gaia, we derive a revised value by
adopting companion and average cluster parallaxes in place of direct Cepheid
parallaxes, and we find H_0 = 72.8 +/- 1.9 (statistical + systematics) +/- 1.9
(ZP) km/s/Mpc when all Cepheids are considered and H0 = 73.0 +/- 1.9
(statistical + systematics) +/- 1.9 (ZP) km/s/Mpc for fundamental mode
pulsators only.
