If the Wiener-Hopf $C^*$-algebra W(G,M) for a discrete group G with a semigroup M has the uniqueness property, then the structure of it is to some extent independent of the choice of isometries on a Hilbert space. In this paper we show that if the Wiener-Hopf $C^*$-algebra W(G,M) of a partially ordered group G with the positive cone M has the uniqueness property, then (G,M) is weakly unperforated. We also prove that the Wiener-Hopf $C^*$-algebra W($mathbb{Z}$, M) of subsemigroup generating the integer group $mathbb{Z}$ is isomorphic to the Toeplitz algebra, but W($mathbb{Z}$, M) does not have the uniqueness property except the case M = $mathbb{N}$.