In this paper we are interested in the study of semi-discrete and full discrete approximations of the null-controllability problem for parabolic equations. We restrict ourselves to the monodimensional case and to finite difference approximations in space. We first show that the semi-discretisation in space of such a problem can be proved to be uniformly controllable with respect to the mesh size if we only try to reach an exponentially small target and not the null target. Then, we extend this result to full discrete problems by using a classical Implicit Euler scheme or a $\theta$-scheme for the time discretization of the problem. The proofs, not given here, are essentially based on the proof of a partial discrete Lebeau-Robbiano inequality which is itself obtained by proving a global Carleman estimate for a semi-discrete elliptic operator. Attractive features of our approach is that it applies to variable coefficient problems and are not restricted to uniform meshes.