We report the detection of continuum and millimeter hydrogen recombination lines (H40$\alpha$, H42$\alpha$ and H50$\beta$) made using ALMA towards the ionized jet associated with the high-mass young stellar object G345.4938+01.4677, the luminous ($L_{\rm bol}\sim70,000~\Lsun$) central dominating source of the high-mass molecular clump associated to IRAS 16562$-$3959. The hydrogen recombination lines exhibit Voigt profiles, a strong signature of the presence of Stark broadening. The pressure broadened Lorentzian line wings imply electron densities of $5\times10^7$~cm$^{-3}$ on average. In addition, we detect SO and SO$_2$ emission arising from a compact ($\sim3000$~AU) molecular structure associated to the central young star. The emission exhibits a velocity gradient perpendicular to the jet-axis, which we interpret as rotation. The set of observations towards G345.4938+01.4677 are consistent with a high-mass associated with a slow photo-ionized wind.
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