The authors investigate solutions of the three dimensional Klein-Gordon and Schrodinger equations in the presence of a new exactly solvable potential of V(r,theta)=-2D(e)(r(e)/r-(1/2)(r(e)(2)/r(2)))+b/r(2) sin(2) theta+a/r(2) cos(2) theta type, the so-called double ring-shaped Kratzer potential. For a diatomic molecule system in double ring-shaped Kratzer potential, the exact bound state energy eigenvalues and corresponding wave functions have been determined within the framework of the asymptotic iteration method. Bound state eigenfunction solutions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric function and Jacobi polynomial. This new formulation is tested by calculating the energies of rovibrational states of a number of diatomic molecules. Also, the author-prove that in the nonrelativistic limit c ->infinity, where c is the speed of light, solutions of the Klein-Gordon system converge to those of the Schrodinger system. (c) 2007 American Institute of Physics.