  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 $L^\infty$
while the initial data is only in $L^\infty$. Moreover, if the memory kernel
depends on a relaxation parameter $\de>0$ and tends to a delta measure weakly
as measures when $\de\to 0+$, then the global entropy solution sequence in $BV$
converges to an admissible solution in $BV$ for the corresponding local
conservation law.
