The present paper deals with a new generalization of Bleimann-Butzer-Hahn operators that depends on a real non-negative parameter α and is therefore called the α-Bleimann-Butzer-Hahn operators. We examined the uniform convergence of the newly defined operators with the help of the Korovkin type approximation theorem. The rate of convergence is investigated by means of the modulus of continuity and by Lipschitz type maximal functions. A Voronovskaya type theorem is also obtained and lastly graphical examples are given in order to illustrate the convergence of the operators to the given functions.