Within the frame of the NUclei of GAlaxies (NUGA) project, we have determined the distribution and kinematics of the molecular gas within the central kpc with high spatial resolution (100-150 pc), for a sample of active galaxies. The goal is to study the gas-fueling mechanisms in AGN. We present interferometric observations of ¹²CO(1-0) and ¹²CO(2-1) line emission from the Seyfert 2 galaxy NGC 6574, obtained with the IRAM Plateau de Bure Interferometer (PdBI). These data have been combined with 30 m mapping data in these lines to correct for the flux resolved by the interferometer. At an angular resolution of 0.7 arcsec (≡110 pc), the ¹²CO(2-1) emission is resolved into an inner disk with a radius of 300 pc. The molecular gas in NGC 6574 is primarily distributed in four components: nucleus, bar, spiral arms - winding up into a pseudo-ring - and an extended underlying disk component. For the overall galaxy host, we find a ¹²CO(2-1) to ¹²CO(1-0) line ratio of ~0.3 indicative of cold or sub-thermally excited gas. For the nucleus, this ratio is close to unity, indicating emission from dense and warm molecular gas. Modeling the gas kinematics with elliptical orbits shows that the molecular gas in the differentially rotating disk of NGC 6574 is strongly influenced by the presence of a stellar bar. The nuclear component shows an extension toward the southeast that may be an indication of the lopsidedness of the nuclear gas distribution. We computed the gravity torques exerted from the stellar bar on the gas, deriving the gravitational potential from near-infrared images, and weighting the torques by the CO distribution. We find negative torques for the gas inside the ring, since the gas aligned with the bar has a slight advance phase shift, leading the bar. This means that gas is flowing in towards the center, at least down to 400 pc in radius, which can explain the observed high nuclear gas concentration. This concentration corresponds to a possible inner Lindblad resonance of the bar, according to the measured rotation curve. The gas has been piling up in this location quite recently, since no startburst has been observed yet.