  We prove that if a unimodular random graph is almost surely planar and has
finite expected degree, then it has a combinatorial embedding into the plane
which is also unimodular. This implies the claim in the title immediately by a
theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding
also implies that all the dichotomy results of [2] about unimodular maps extend
in the one-ended case to unimodular random planar graphs.
