  This work is part of the NUGA survey of CO emission in nearby active
galaxies. We present observations of NGC4569, a member of the Virgo Cluster. We
analyse the molecular gas distribution and kinematics in the central region and
we investigate a possible link to the strong starburst present at the nucleus.
70% of the 1.1x10^9 Msolar of molecular gas detected in the inner 20" is found
to be concentrated within the inner 800 pc and is distributed along the large
scale stellar bar seen in near-infrared observations. A hole in the CO
distribution coincides with the nucleus where most of the Halpha emission and
blue light are emitted. The kinematics are modelled in three different ways,
ranging from the purely geometrical to the most physical. This approach allows
us to constrain progressively the physical properties of the galaxy and
eventually to emerge with a reasonable fit to an analytical model of orbits in
a barred potential. Fitting an axisymmetric model shows that the non-circular
motions must be comparable in amplitude to the circular motions (120 km/s).
Fitting a model based on elliptical orbits allows us to identify with
confidence the single inner Lindblad resonance (ILR) of the large scale bar.
Finally, a model based on analytical solutions for the gas particle orbits in a
weakly barred potential constrained by the ILR radius reproduces the
observations well. The mass inflow rate is then estimated and discussed based
on the best fit model solution. The gravitational torques implied by this model
are able to efficiently funnel the gas inside the ILR down to 300 pc, although
another mechanism must take over to fuel the nuclear starburst inside 100 pc.
