We present a bijection between balanced Delannoy paths of length 2n and the faces of the n-dimensional Simion type B associahedron. This polytope is also known as the Bott-Taubes polytope and the cyclohedron. This bijection takes a path with k up steps (and k down steps) to a (k-1)-dimensional face of the Simion type B associahedron. We give two presentations of this bijection, one recursive and one non-recursive.