  Brownian motion with darning (BMD in abbreviation) is introduced and studied
in [4] and [5, Chapter 7]. Roughly speaking, BMD travels across the "darning
area" at infinite speed, while it behaves like a regular BM outside of this
area. In this paper we show that starting from a single point in its state
space, BMD is the weak limit of a family of continuous-time simple random walks
on square lattices with diminishing mesh sizes. From any vertex in their state
spaces, the approximating random walks jump to its nearest neighbors with equal
probability after an exponential holding time.
