We consider the problem of designing robust numerical integration scheme of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as x(alpha), with alpha > 1. We propose an (semi-explicit) exponential-Euler scheme for which we obtain a theoretical convergence rate for the weak error. To this aim, we analyze the C-1,C-4 regularity of the solution of the associated backward Kolmogorov PDE using its Feynman-Kac representation and the flow derivative of the involved processes. Under some suitable hypotheses on the parameters of the model, we prove a rate of weak convergence of order one for the proposed exponential Euler scheme, and illustrate it with some numerical experiments.