The X-ray transient Swift J1727.8‑1613 ended its 10-month discovery outburst in June of 2024, when it reached an optical brightness comparable to pre-discovery magnitudes. With the aim of performing a dynamical study, we launched an optical spectroscopy campaign with the GTC telescope. We detected the companion star and constructed its radial velocity curve, yielding a binary orbital period of Porb = 10.8038 ± 0.0010 h and a radial velocity semi-amplitude of K2 = 390 ± 4 km s‑1. This results in a mass function of f(M1) = 2.77 ± 0.09 M⊙. Combined with constraints on the binary inclination, it sets a lower limit on the compact object mass of M1 > 3.12 ± 0.10 M⊙, dynamically confirming the black hole nature of the accretor. A comparison of the average spectrum in the rest frame of the companion with synthetic stellar templates supports a K4V donor that is partially (74%) veiled by the accretion disc. A refined distance measurement of 3.4 ± 0.3 kpc, together with the astrometric proper motion and the systemic velocity derived from the radial velocity curve (γ = 181 ± 4 km s‑1), supports a natal kick velocity of vkick = 210‑50+40 km s‑1, at the upper end of the observed distribution.