Very recently, in  Chen et. al introduced and considered a new generalization of Bernstein polynomials depending on a patameter alpha. As classical Bernstein operators, these operators also provide interpolation at the end points of [0,1] and they have the linear precision property which means those reproduce the linear functions. In this paper we introduce genuine alpha-Bernstein-Durrmeyer operators. Some approximation results, which include local approximation, error estimation in terms of Ditzian-Totik modulus of smoothness are obtained. Also, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.