In this paper we prove that the null controllability property of the heat equation may beobtained as limit of the exact controllability properties of singularly perturbed damped wave equations. We impose Dirichlet, homogeneous boundary conditions. The control is supported in a neighborhood ofa subset of the boundary that satisfies the classical requirements to apply multiplier techniques. The proofneeds an iterative argument that allows to treat separately the low and high frequencies and to make use ofthe dissipativity of the systems under consideration. This is combined with sharp observability estimates onthe eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano, and global Carleman estimates.This proof applies in any space dimension.As a consequence of the uniform controllability we derive uniform observability estimates which cannot be proved by classical methods due to the singular character of the perturbations we deal with.