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We report ALMA observations of CO(3-2) emission in the Seyfert 1 galaxy NGC1566, at a spatial resolution of 25 pc. Our aim is to investigate themorphology and dynamics of the gas inside the central kpc, and to probe nuclearfueling and feedback phenomena. NGC 1566 has a nuclear bar of 1.7 kpc radiusand a conspicuous grand design spiral starting from this radius. The ALMA fieldof view, of diameter 0.9 kpc, lies well inside the nuclear bar and reveals amolecular trailing spiral structure from 50 to 300~pc in size, which iscontributing to fuel the nucleus, according to its negative gravity torques.  The spiral starts with a large pitch angle from the center and then winds upin a pseudo-ring at the inner Lindblad resonance (ILR) of the nuclear bar.  This is the first time that a trailing spiral structure is clearly seendriving the gas inwards inside the ILR ring of the nuclear bar. This phenomenonshows that the massive central black hole has a significant dynamical influenceon the gas, triggering its fueling.  The gaseous spiral is well correlated with the dusty spiral seen throughextinction in HST images, and also with a spiral feature emitting 0.87mmcontinuum. This continuum emission must come essentially from cold dust heatedby the interstellar radiation field. The HCN(4-3) and HCO+(4-3) lines weresimultaneously mapped and detected in the nuclear spiral. The HCO+(4-3) line is3 times stronger than the HCN(4-3), as expected when star formation excitationdominates over active galactic nucleus (AGN) heating. The CO(3-2)/HCO+(4-3)integrated intensity ratio is \sim 100.  The molecular gas is in remarkably regular rotation, with only slightnon-circular motions at the periphery of the nuclear spiral arms. Theseperturbations are quite small, and no outflow nor AGN feedback is detected.

ALMA reveals the feeding of the Seyfert 1 nucleus in NGC 1566