We report on the first ALMA observation of the CO(3−2) and rest-frame ∼340 GHz continuum emission in PDS 456, which is the most luminous, radio-quiet QSO in the local Universe (z ≃ 0.18), with a bolometric luminosity L_Bol ∼ 10⁴⁷ erg s⁻¹. ALMA angular resolution allowed us to map scales as small as ∼700 pc. The molecular gas reservoir traced by the core of the very bright CO(3−2) emission line is distributed in a compact rotating disk, with a size of ∼1.3 kpc, seen close to face-on (i ∼ 25 deg). Fast CO(3−2) emission in the velocity range v ∈ [ − 1000, 500] km s⁻¹ is also present. Specifically, we detect several blue-shifted clumps out to ∼5 kpc from the nucleus, in addition to a compact (R ≲ 1.2 kpc), broad emission component. These components reveal a galaxy-wide molecular outflow, with a total mass M_mol^out ∼ 2.5 × 10⁸ M_⊙ (for an α_CO = 0.8 M_⊙ (K km s⁻¹ pc²)⁻¹) and a mass outflow rate Ṁ_mol ∼ 290 M_⊙ yr⁻¹. The corresponding depletion time is τ_dep ∼ 8 Myr, shorter than the rate at which the molecular gas is converted into stars, indicating that the detected outflow is potentially able to quench star-formation in the host. The momentum flux of the molecular outflow normalised to the radiative momentum output (i.e. L_Bol/c) is ≲1, comparable to that of the X-ray ultra-fast outflow (UFO) detected in PDS 456. This is at odds with the expectations for an energy-conserving expansion suggested for most of the large-scale outflows detected in low-luminosity AGNs so far. We suggest three possible scenarios that may explain this observation: (i) in very luminous AGNs such as our target the molecular gas phase is tracing only a fraction of the total outflowing mass; (ii) a small coupling between the shocked gas by the UFO and the host-galaxy interstellar medium (ISM); and (iii) AGN radiation pressure may be playing an important role in driving the outflow.