  A fully discrete and fully explicit low-regularity integrator is constructed
for the one-dimensional periodic cubic nonlinear Schr\"odinger equation. The
method can be implemented by using fast Fourier transform with $O(N\ln N)$
operations at every time level, and is proved to have an $L^2$-norm error bound
of $O(\tau\sqrt{\ln(1/\tau)}+N^{-1})$ for $H^1$ initial data, without requiring
any CFL condition, where $\tau$ and $N$ denote the temporal stepsize and the
degree of freedoms in the spatial discretisation, respectively.
