We present evidence that the peculiar interacting starburst galaxy NGC 2782 (Arp 215) harbors a gas-rich nuclear stellar bar feeding an M82-class powerful central starburst, from a study based on OVRO CO (J=1->0) data, WIYN BVR & Halpha observations, along with available NIR images, a 5 GHz RC map and HST images. NGC 2782 harbors a clumpy, bar-like CO feature of radius ~ 7.5" (1.3 kpc) which leads a nuclear stellar bar of similar size. The nuclear CO bar is massive: it contains ~2.5x10**9 M_sun

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... of molecular gas, which makes up ~ 8 % of the dynamical'mass present within a 1.3 kpc radius. Within the CO bar, emission peaks in two extended clumpy lobes which lie on opposite sides of the nucleus, separated by ~ 6" (1 kpc). Between the CO lobes, in the inner 200 pc radius, resides a powerful central starburst which is forming stars at a rate of 3 to 6 M_sun yr-1. While circular motions dominate the CO velocity field, the CO lobes show weak bar-like streaming motions on the leading side of the nuclear stellar bar, suggestive of gas inflow. We estimate semi-analytically the gravitational torque from the nuclear stellar bar on the gas, and suggest large gas inflow rates from the CO lobes into the central starburst. These observations, which are amongst the first ones showing a nuclear stellar bar fuelling molecular gas into an intense central starburst, are consistent with simulations and theory which suggest that nuclear bars provide an efficient way of transporting gas closer to the galactic center to fuel central activity. Furthermore, several massive clumps are present at low radii, and dynamical friction might produce further gas inflow. We suggest that the nuclear molecular gas bas and central activity will be very short-lived, likely disappearing within 5x10**8 years.