We consider a fractal scalar conservation law, that is to say a conservation law modified by a fractional power of the Laplace operator, and we propose a numerical method to approximate its solutions. We make a theoretical study of the method, proving in the case of an initial data belonging to L∞ ∩ BV that the approximate solutions converge in L∞ weak-∗ and in Lp strong for p < ∞, and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law.