In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order α in the nonlinear term and a β-fractional Laplacian; we consider the critical case α+β=[formula presented] and we assume [formula presented]<β<[formula presented]. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order α. We prove global well posedness when the initial velocity is in Hr and the initial temperature is in Hr−β for r>max(2β,β+1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.