The authors obtain a new property of the n-dimensional binary undirected de Bruijn graph UB(n) for n≥4, namely, there is a vertex x such that for any other vertex y there exist at least two internally disjoint paths of length at most n-1 between x and y in UB(n). The result means that the (n-1,2)-dominating number of UB(n) is equal to one if n≥4.