This paper presents a new tensor completion method named minimum volume constraint tensor completion. Unlike the nuclear norm penalization based methods, our method extends the conception of the matrix volume to the tensor volume, and uses the volume measurement as the penalization to address the tensor completion problem. The alternating direction method of multipliers (ADMM) algorithm is then employed to solve the optimization problem of the proposed model. Experimental results on several popular databases show superior performance of our method compared to the nuclear norm penalization based methods in terms of the accuracy and robustness.