The existence of the large-amplitude dust inertial Alfvén waves (DIAWs) has been presented in an electron-depleted, two-fluid dust-ion plasma. Linear analysis shows that the DIAWs travel slower than the dust Alfvén waves. DIAWs are the obliquely (with respect to the external magnetic field) propagating oscillations ofdust density, having the characteristics of a solitary wave. In order to observe the nonlinear behaviour of the DIAWs, the Sagdeev pseudopotential method has been used to derive the energy balance equation and from the expressionof the Sagdeev pseudopotential, the existence conditions for the DIAWs have also been determined. It is observed that density rarefactions travelling at sub- and super-Alfvénic speeds are associated with DIAWs.