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.
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