In this paper, the exp-function method is generalized to solve a first-order nonlinear differential equation with forth-degree nonlinear term. As a result, three new exact solutions with free parameters are obtained, the first two of which include generalized hyperbolic function solutions and generalized trigonometric solutions as special cases and the third one is a rational solution. It is shown that the generalized exp-function method with the help of computer symbolic computation provides a straightforward and effective mathematical tool for solving nonlinear differential equations.