In this article, we consider the null controllability problem for the cubic semi-linear heat equation in bounded domains Ω of R n , n ≥ 3 with Dirichlet boundary conditions for small initial data. A constructive way to compute a control function acting on any nonempty open subset ω of Ω is given such that the corresponding solution of the cubic semilinear heat equation can be driven to zero at a given final time T. Furthermore, we provide a quantitative estimate for the smallness of the size of the initial data with respect to T that ensures the null controllability property.