In this work we characterize the smooth surfaces that can be embedded in the Grassmannian of lines $G(1,5)$ with one apparent double point, i.e., such that the general linear projection to $G(1,3)$ produces just one double point. The result is that such a surface must be either a rational scroll of degree $4$ or $5$ or a Del Pezzo surface of degree $6$ or $7$.
