We present a new potential function with an exactly solvable Ｓｃｈｒ?ｄｉｎｇｅｒ equation. At the same time, we propose a new variable transformation to show that the hyperbolic Ｐ?ｓｃｈｌ-Ｔｅｌｅｒ molecular potential is also exactly solvable. The form of the solutions is not only closed, but also very simple. The normalized wave functions are expressed by the hypergeometric series. All the results of the Hulthéｎ potential and the modified Ｐ?ｓｃｈｌ-Ｔｅｌｅｒ potential as well as the reflectionless potential in the existing literature are contained in the more general conclusions of this paper as special cases. Finally, practical applications of the potentials in this paper to diatomic molecules are discussed.