We present the metallicity and radial velocity for 450 bona fide members of the Sagittarius dwarf spheroidal (Sgr dSph) galaxy, measured from high-resolution spectra (R ≃ 18 000) obtained with FLAMES at the VLT. The targets were carefully selected (a) to sample the core of the main body of Sgr dSph while avoiding contamination from the central stellar nucleus, and (b) to prevent any bias on the metallicity distribution by selecting targets based on their Gaia parallax and proper motions. All the targets selected in this way were confirmed as radial velocity members. We used this sample to derive the first metallicity distribution of the core of Sgr dSph, which is virtually unaffected by metallicity biases. The observed distribution ranges from [Fe/H] ≃ −2.3 to [Fe/H] ≃ 0.0, with a strong, symmetric, and relatively narrow peak around [Fe/H] ≃ −0.5 and a weak and extended metal-poor tail, in which only 13.8 ± 1.9% of the stars have [Fe/H] < −1.0. We confirm previous evidence of correlations between chemical and kinematical properties of stars in the core of Sgr. In our sample, stars with [Fe/H] ≥ −0.6 display a lower velocity dispersion and a higher rotation amplitude than those with [Fe/H] < −0.6, confirming previous suggestions of disk/halo structure for the progenitor of the system.