  We report ALMA observations of CO(3-2) emission in the Seyfert 1 galaxy NGC
1566, at a spatial resolution of 25 pc. Our aim is to investigate the
morphology and dynamics of the gas inside the central kpc, and to probe nuclear
fueling and feedback phenomena. NGC 1566 has a nuclear bar of 1.7 kpc radius
and a conspicuous grand design spiral starting from this radius. The ALMA field
of view, of diameter 0.9 kpc, lies well inside the nuclear bar and reveals a
molecular trailing spiral structure from 50 to 300~pc in size, which is
contributing to fuel the nucleus, according to its negative gravity torques.
  The spiral starts with a large pitch angle from the center and then winds up
in 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 seen
driving the gas inwards inside the ILR ring of the nuclear bar. This phenomenon
shows that the massive central black hole has a significant dynamical influence
on the gas, triggering its fueling.
  The gaseous spiral is well correlated with the dusty spiral seen through
extinction in HST images, and also with a spiral feature emitting 0.87mm
continuum. This continuum emission must come essentially from cold dust heated
by the interstellar radiation field. The HCN(4-3) and HCO+(4-3) lines were
simultaneously mapped and detected in the nuclear spiral. The HCO+(4-3) line is
3 times stronger than the HCN(4-3), as expected when star formation excitation
dominates 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 slight
non-circular motions at the periphery of the nuclear spiral arms. These
perturbations are quite small, and no outflow nor AGN feedback is detected.
