The spectral classifications of the Galactic O-Star Spectroscopic Survey (GOSSS) and the astrometric and photometric data from Gaia have significantly improved our ability to measure distances and determine memberships of stellar groups (clusters, associations, or parts thereof) with OB stars. In the near future, the situation will be further improved thanks to subsequent Gaia data releases and new photometric and spectroscopic surveys. We initiated a program to identify and determine the membership of Galactic stellar groups with OB stars and measure distances to them. Given the data currently available, we started with the identification and distance determinations of groups with O stars. In this paper, we concentrate on groups that contain stars with the earliest spectral subtypes. We used GOSSS to select Galactic stellar groups with O2-O3.5 stars and the method described in paper 0 of this series, which combines Gaia DR2 G + G_BP_ + G_RP_ photometry, positions, proper motions, and parallaxes to assign robust memberships and measure distances. We also included Collinder 419 and NGC 2264, the clusters cited in that paper, to generate our first list of 16 O-type Galactic stellar groups. We derived distances, determined the membership, and analyzed the structure of sixteen Galactic stellar groups with O stars, Villafranca O-001 to Villafranca O-016, including the fourteen groups with the earliest-O-type optically accessible stars known in the Milky Way. We compared our distance with previous results from the literature and establish that the best consistency is with (the small number of) VLBI parallaxes and the worst is with kinematic distances. Our results indicate that very massive stars can form in relatively low-mass clusters or even in near-isolation, as is the case for the Bajamar star in the North America nebula. This lends support to the hierarchical scenario of star formation, where some stars are born in well-defined bound clusters but others are born in associations that are unbound from the beginning: groups of newborn stars come in many shapes and sizes. We propose that HD 64568 and HD 64315 AB could have been ejected simultaneously from Haffner 18 (Villafranca O-012 S). Our results are consistent with a difference of =~20{mu}as in the Gaia DR2 parallax zero point between bright and faint stars.