- We present a detailed strong lensing (SL) mass reconstruction of the core of the galaxy cluster MACS J2129.4−0741 (z_(cl) = 0.589) obtained by combining high-resolution Hubble Space Telescope photometry from the CLASH (Cluster Lensing And Supernovae survey with Hubble) survey with new spectroscopic observations from the CLASH-VLT (Very Large Telescope) survey. A background bright red passive galaxy at z_(sp) = 1.36, sextuply lensed in the cluster core, has four radial lensed images located over the three central cluster members. Further 19 background lensed galaxies are spectroscopically confirmed by our VLT survey, including 3 additional multiple systems. A total of 31 multiple images are used in the lensing analysis. This allows us to trace with high precision the total mass profile of the cluster in its very inner region (R < 100 kpc). Our final lensing mass model reproduces the multiple images systems identified in the cluster core with high accuracy of 0.4 arcsec. This translates to a high-precision mass reconstruction of MACS 2129, which is constrained at a level of 2 per cent. The cluster has Einstein parameter Θ_E = (29 ± 4) arcsec and a projected total mass of M_(tot)(<Θ_E) = (1.35 ± 0.03) × 10^(14) M⊙ within such radius. Together with the cluster mass profile, we provide here also the complete spectroscopic data set for the cluster members and lensed images measured with VLT/Visible Multi-Object Spectrograph within the CLASH-VLT survey.