As part of a radial velocity survey of low Galactic latitude structures that we undertook with the 2dF spectrograph on the AAT, we present the radial velocities of more than 1500 Red Giant Branch and Red Clump stars towards the centre of the Canis Major dwarf galaxy. With a mean velocity of 72pm7 km/s at a Heliocentric distance of 5.5 kpc and 114pm2 km/s at 8.5 kpc, these stars present a peculiar distance -- radial velocity relation that is unlike that expected from thin or thick disc stars. Moreover, they belong to a kinematically cold population with an intrinsic dispersion that may be as low as 11_{-1}^{+3} km/s. The radial velocity distribution is used to select Canis Major stars in the UCAC2.0 proper motion catalogue and derive proper motions in Galactic coordinates of (mu_l,mu_b)= (-3.6pm0.8 mas/yr, 1.5pm0.4 mas/yr) for the dwarf galaxy, which after correcting for the reflex solar motion along this line-of-sight gives (mu_l,mu_b)= (-6.8pm0.8 mas/yr, 0.8pm0.4 mas/yr)$, corresponding to a prograde orbit with a tangential velocity of sim235 km/s at the average distance of sim7.2 kpc. All these kinematic constraints can be reproduced in simulations of the accretion of a dwarf onto the Galactic disc. Such a process could also be responsible for the Monoceros Ring that has recently been shown to encompass the Galactic disc. However, without constraints on the kinematics of the tidal arms emerging from the Canis Major dwarf, it is not yet possible to definitively prove a link between the two structures.
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