Aims. High-resolution Very-Long-Baseline Interferometry observations of relativistic jets are essential for constraining the fundamental parameters of jet formation models. At a distance of 249 Mpc, Cygnus A is a unique target for such studies, since it is the only Fanaroff-Riley Class II radio galaxy for which a detailed subparsec scale imaging of the base of both jet and counter-jet can be obtained. Observing at millimeter wavelengths unveils those regions that appear self-absorbed at longer wavelengths and enables an extremely sharp view toward the nucleus to be obtained. Methods. We performed 7 mm Global VLBI observations, achieving ultra-high resolution imaging on scales down to 90uas. This resolution corresponds to a linear scale of only ~400 Schwarzschild radii (for MBH = 2.5 × 109M⊙). We studied the kinematic properties of the main emission features of the two-sided flow and probed its transverse structure through a pixel-based analysis. Results. We suggest that a fast and a slow layer with different acceleration gradients exist in the flow. The extension of the acceleration region is large (~ 104RS), indicating that the jet is magnetically driven. The limb brightening of both jet and counter-jet and their large opening angles (φJ ~ 10°) strongly favour a spine-sheath structure. In the acceleration zone, the flow has a parabolic shape (r ∝ z0.55 ± 0.07). The acceleration gradients and the collimation profile are consistent with the expectations for a jet in "equilibrium", achieved in the presence of a mild gradient of the external pressure (p ∝ z− k,k ≤ 2).