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In this paper, we characterize the $n$-dimensional $(n\ge 3)$ complete spacelike hypersurfaces $M^n$ in a de Sitter space $S_1^{n+1}$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^n$ is a locus of moving $(n-1)$-dimensional submanifold $M_1^{n-1}\left(s\right)$, along $M_1^{n-1}\left(s\right)$ the principal curvature $\lambda$ of multiplicity $n-1$ is constant and $M_1^{n-1}\left(s\right)$ is umbilical in $S_1^{n+1}$ and is contained in an $(n-1)$-dimensional sphere $S^{n-1}\left(c\left(s\right)\right)=E^n\left(s\right)\cap S_1^{n+1}$ and is of constant curvature ${\left(\frac{d\{log|{\lambda }^2-(1-R){|}^{1/n}\}}{ds}\right)}^2-{\lambda }^2+1$,where $s$ is the arc length of an orthogonal trajectory of the family...