We have studied with the IRAM interferometer the (CO)-C-12(1-0) emission around the nucleus of NGC 5907, an edge-on spiral at the center of which we had previously detected fast rotating molecular gas. The high angular resolution (3.1''>, or 160 pc along the major axis) allowed an accurate determination of NGC 5907's kinematical center (alpha(J1950.0) = 15(h)14(m)35.5(s), delta(J1950.0)= 56 degrees 30' 43.3''). Within +/-2'' from the center, we detect molecular gas with velocities of +/-160 kms(-1) (relative to the source systemic velocity V-sys = 677 kms(-1) (LSR)), and, within +/-10'' from the center, velocities of up to +/-180 km/s. Part of the gas exhibits non-circular motions. We explain the gas kinematics by the presence of a stellar bar with an angular velocity (Ohm(p) = 70 kmskpc(-1)) and of an Outer Inner Linblad Resonance (oILR) at r similar or equal to 14''. The dynamical mass inside the R < 300 pc region, estimated from the steepness of the apparent rotation curve, is similar or equal to 5 10(9)M.. It is comparable to the stellar mass derived from near IR photometry and > 10 times larger than the mass of gas in the same region. The CO brightness distribution and, presumably, the molecular gas distribution are highly asymmetrical. The interferometer, which filters out the smooth disk emission detected by the 30-m telescope, reveals two narrow structures parallel to the major axis. We identify those with spiral arms. From their position and velocities, we determine an inclination i = 86.5 degrees of the plane of the galaxy. The thickness of the arms, perpendicularly to this plane, is found to be less than or equal to 3'' (less than or equal to 150 pc). The large mass concentration near the nucleus and the flat rotation curve of NGC 5907 are more typical of early or intermediate-type spirals than of late type spirals. Yet, the large CO and HI luminosties, the conspicuous molecular arms, and the absence of visible bulge on optical photographs are more characteristic of Sc-Sd galaxies. NGC5907 may thus be at a critical stage of galactic evolution.