Extreme helium stars (EHe stars) are hydrogen-deficient supergiants of spectral type A and B. They are believed to result from mergers in double degenerate systems. In this paper we present a detailed quantitative non-LTE spectral analysis for BD+10$^circ$2179, a prototype of this rare class of stars, using UVES and FEROS spectra covering the range from $sim$3100 to 10 000 {AA}. Atmosphere model computations were improved in two ways. First, since the UV metal line blanketing has a strong impact on the temperature-density stratification, we used the Atlas12 code. Additionally, We tested Atlas12 against the benchmark code Sterne3, and found only small differences in the temperature and density stratifications, and good agreement with the spectral energy distributions. Second, 12 chemical species were treated in non-LTE. Pronounced non-LTE effects occur in individual spectral lines but, for the majority, the effects are moderate to small. The spectroscopic parameters give $T_mathrm{eff}$ = 17 300$pm$300 K and $log g$ = 2.80$pm$0.10, and an evolutionary mass of 0.55$pm$0.05 $M_odot$. The star is thus slightly hotter, more compact and less massive than found in previous studies. The kinematic properties imply a thick-disk membership, which is consistent with the metallicity $[$Fe/H$]approx-1$ and $alpha$-enhancement. The refined light-element abundances are consistent with the white dwarf merger scenario. We further discuss the observed helium spectrum in an appendix, detecting dipole-allowed transitions from about 150 multiplets plus the most comprehensive set of known/predicted isolated forbidden components to date. Moreover, a so far unreported series of pronounced forbidden He I components is detected in the optical-UV.
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