Abstract: Global entropy solutions in BV for a scalar nonlocal conservation law with fading memory are constructed as limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in BV are established, which also yield the existence of entropy solutions in L^∞ while the initial data is only in L^∞. Moreover, if the memory kernel depends on a relaxation parameter ε>0 and tends to a delta measure weakly as measures when ε→0+, then the global entropy solution sequence in BV converges to an admissible solution in BV for the corresponding local conservation law.