This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor $𝓐_τ$ in H¹(ℝ¹). In addition, we show that the global attractor $𝓐_τ$ is regular, i.e., $𝓐_τ$ is actually included, bounded and compact in H²(ℝ¹). Finally, we estimate the finite fractal dimensions of $𝓐_τ$.