In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ${\partial y\over\partial n} + \beta\,y = 0$. We consider distributed controls with support in a small set and nonregular coefficients $\beta=\beta(x,t)$. For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.