The peculiar Sbc-type starburst galaxy NGC 3310 was mapped in the 21 cm H I line at Westerbork with an angular resolution of 12''.2 x 15''.1 (alpha x delta) and a velocity resolution of 33 km s(-1). The galaxy was also partially mapped in the CO(1-0) and (2-1) lines at IRAM with resolutions of similar to 23'' and similar to 12'', respectively. In the H I line, an extension of the optical jet-like feature in the Northwestern outer parts (the 'arrow') is observed, extending outward to similar to 6'.5 (38 kpc) from the centre in data degraded to a resolution of 60''. A region of H I emission is found some 9' to the South of NGC 3310 at radial velocities around 1040 km s(-1), i.e. just above systemic, which may be connected to the 'main body' H I by a weak bridge. At higher resolutions, the H I shows a central 'hole' and two ridges partially coincident with the inner arms seen in H alpha and continuing outwards as an extension of these, bending strongly just beyond their optical extent. The central hole is centered on the Southeastern side of the optical inner ring, not on the optical nucleus. The Northern ridge seems to connect its H alpha counterpart in the inner regions to the 'arrow' in the outer parts. Going from the optical disc to the outer regions, radial velocities measured in the H I gas generally change gradually, with the notable exception of the region of the optical 'arrow'. The receding half of the galaxy shows rather normal kinematics, as do the inner similar to 70'' of radius of the half with velocities below systemic. Severely disturbed H I kinematics RTP found in the approaching half beyond that radius, which corresponds to the domain of the optical 'bow and arrow' feature. Fits of a circular rotation model to the apparently undisturbed parts of the observed H I velocity field yield a kinematic inclination of 52 degrees +/- 2 degrees and major axis position angle of 167 degrees +/- 1 degrees. Subtraction of a model velocity field based on these results from that observed in H I confirms that the gas in the Northwestern (disturbed) half of the galaxy exhibits highly non-circular motions, with radial velocity deviations up to 120 km s(-1). The observed H I kinematics are most naturally explained as due to a recent merger of NGC 3310 with a gas-rich dwarf galaxy. While being quite normal in its optical and H I global properties, NGC 3310 has an exceptionally low rotation velocity at radii near its de Vaucouleurs' radius (similar to 80 km s(-1)), and hence a very low dynamical mass inside that radius (similar to 2 10(10) M.). This may indicate that the pre-merger object was a dwarf spiral galaxy. The CO observations show that NGC 3310 has rather weak CO emission, implying a low H-2 mass if a standard conversion is appropriate. If so, the ratio of H-2 to H I gas mass is typical for a very late-type spiral, once more suggesting a recent merger with such an object. The ratio of CO(2-1) to CO(1-0) emission is exceptionally high. This may be due to the CO(1-0) line being optically thin. Combined with a low metallicity, a strong UV radiation field and a high dust temperature in the interstellar medium, this probably results in a CO flux to molecular hydrogen column density conversion factor lower than Galactic, closer to that found in the Magellanic clouds