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

Unimodular random one-ended planar graphs are sofic