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A map is called edge-regular if it is edge-transitive but not arc-transitive. In this paper, we show that a complete graph Kn{K}_{n} has an orientable edge-regular embedding if and only if n=pd>3n={p}^{d}\gt 3 with p an odd prime such that pd≡3{p}^{d}\equiv 3 (mod4)(\mathrm{mod}\hspace{.25em}4). Furthermore, Kpd{K}_{{p}^{d}} has pd−34dϕ(pd−12)\tfrac{{p}^{d}-3}{4d}\hspace{0.25em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) non-isomorphic orientable edge-regular embeddings.