  We consider a discrete nonlinear Schr\"odinger equation with memory effect on
the lattice $h\mathbb{Z}$ with mesh-size $h>0$. As $h\rightarrow 0$, we prove
that solutions to this discrete equation converge strongly in $L^2$ to the
solution to a continuous NLS-type equation with a memory effect, and compute
the precise rate of convergence. In the process, we extend some recent ideas
proposed by Hong and Yang in order to exploit a smoothing effect. This approach
could therefore be adapted to tackle continuum limits of more general
dispersive equations that require working in similar spaces.
