This paper presents the development of two-point diagonally implicit fractional block backward differentiation formula of order two with constant step size (2DIFBBDF(2)) for solving the fractional differential equations (FDEs). The method is derived based on the concept of fractional linear multistep method and the classical diagonally block backward differentiation formula (BBDF) method. Convergence and stability analyses of the method are also discussed. This method is proved to be A-stable for values of fractional order between 0.7 and 1.0. Next, numerical examples in the form of linear, non-linear and system of FDEs are presented to demonstrate the method's reliability and efficiency. The results obtained are compared with the existing methods whereas 2DIFBBDF(2) method outperforms the others in terms of accuracy indicating that it is an appropriate method for solving FDEs.