  Little is known about how high-mass stars form. Around 30\% of the young
high-mass stars in the Galaxy are observed at the edges of ionized (HII)
regions. Therefore these are places of choice to study the earliest stages of
high-mass star formation, especially towards the most massive condensations.
High-spatial resolution observations in the millimeter range might reveal how
these stars form and how they assemble their mass. We want to study the
fragmentation process down to the 0.01~pc scale in the most massive
condensation observed at the south-western edge of the HII region RCW~120 where
the most massive Herschel cores ($\sim$124~$M_{\odot}$ in average) could form
high-mass stars. Using ALMA 3~mm continuum observations towards the densest and
most massive millimetric condensation (Condensation 1) of RCW~120, we used the
getimages and getsources algorithms to extract the sources detected with ALMA
and obtained their physical parameters. The fragmentation of the hersche cores
is discussed through their Jeans mass to understand the properties of the
future stars. We extracted 18 fragments from the ALMA continuum observation at
3~mm towards 8 cores detected with Herschel, whose mass and deconvolved size
range from 2~$M_{\odot}$ to 32~$M_{\odot}$ and from 1.6~mpc to 28.8~mpc,
respectively. The low degree of fragmentation observed, regarding to the
thermal Jeans fragmentation, suggests that the observed fragmentation is
inconsistent with ideal gravitational fragmentation and other ingredients such
as turbulence or magnetic fields should be added in order to explain it.
Finally, the range of fragments' mass indicates that the densest condensation
of RCW~120 is a favourable place for the formation of high-mass stars with the
presence of a probable UCHII region associated with the 27~$M_{\odot}$ Fragment
1 of Core 2.
