Summary: The convection diffusion equation exists widely in many fields. In order to solve some practical problems, the discretization scheme should not only satisfy some basic properties, such as convergence, stability and the existence and uniqueness of solutions, but also keep the positivity of the discretization scheme. A lot of researches has been done to solve the convection diffusion equation by using the finite volume scheme, but little work has been done in the aspect of keeping the positivity. In this paper, a nonlinear positivity-preserving finite volume scheme for the one-dimensional convection diffusion equation on arbitrary nonuniform grids is constructed. The scheme is unformed in a matrix form. Then, it is proved that the scheme satisfies the requirement of positivity-preserving by using the properties of the coefficient matrix. The scheme only contains the unknown quantity of the center of the interval element and satisfies the local conservation of flux at the end of the interval. Finally, the numerical results show that the proposed scheme is effective and owns the second order accuracy. In addition, the scheme is applicable to the solution of problems with discontinuous diffusion coefficients.