Let X 1 , X 2 , . . . be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the formWe show, under mild assumptions on the law of X i , that one can choose the scale factor a N in such a way that the process (Y N ⌊N t⌋ ) t∈R + converges in distribution to a given diffusion (M t ) t∈R + solving a stochastic differential equation with possibly irregular coefficients, as N → ∞. To this end we embed the scaled random walks into the diffusion M with a sequence of stopping times with expected time step 1/N. 2010 MSC : 60F17, 60J60, 65C30. Keywords : stochastic differential equations, irregular diffusion coefficient, weak law of large numbers for u.i. arrays, weak convergence of processes, Skorokhod embedding problem.