We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from zero, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.