In general, one of the dual boundary integral equations (BIE) is always a kind of derivative BIE with the hypersingular integral. The treatment of the hyper-singularities is difficult in the boundary element methods (BEM). This paper focuses on the BIE in the two-dimensional potential problems. A series of transformations are manipulated on the conventional potential derivative BIE in order to eliminate the hyper-singularity. It leads to a new natural BIE in the two-dimensional potential problems. The natural BIE also belongs to the derivative BIE, but only contains the strongly singular integral. The evaluation for the strongly singular integral is given by the subtraction method. As a result, another boundary element analysis according to the natural BIE can obtain more accurate potential derivatives on the boundary in comparison with the conventional BEM. Furthermore, the natural BIE can also be applied to calculating the potential derivatives at the interior points very close to the boundary. Some comparisons with exact solutions are done to illustrate the application and efficiency of the natural BIE.