A certain two-state exponential potential model is solved quantum mechanically exactly. Compact expressions for nonadibatic transition matrices are obtained. Interesting quantum mechanical threshold effects are found. Simple very accurate expressions are found from a semiclassical viewpoint for the nonadiabatic transition probabilities, indicating that the exponential model may present a third important basic model in addition to the Landau-Zener-Stueckelberg and the Rosen-Zener-Demkov models. Extension to general cases is also briefly discussed.